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Theorem hartogslem1 9534
Description: Lemma for hartogs 9536. (Contributed by Mario Carneiro, 14-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
Hypotheses
Ref Expression
hartogslem.2 𝐹 = {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))}
hartogslem.3 𝑅 = {βŸ¨π‘ , π‘‘βŸ© ∣ βˆƒπ‘€ ∈ 𝑦 βˆƒπ‘§ ∈ 𝑦 ((𝑠 = (π‘“β€˜π‘€) ∧ 𝑑 = (π‘“β€˜π‘§)) ∧ 𝑀 E 𝑧)}
Assertion
Ref Expression
hartogslem1 (dom 𝐹 βŠ† 𝒫 (𝐴 Γ— 𝐴) ∧ Fun 𝐹 ∧ (𝐴 ∈ 𝑉 β†’ ran 𝐹 = {π‘₯ ∈ On ∣ π‘₯ β‰Ό 𝐴}))
Distinct variable groups:   𝑓,𝑠,𝑑,𝑀,𝑦,𝑧   𝑓,π‘Ÿ,π‘₯,𝐴,𝑦   𝑅,π‘Ÿ,π‘₯   𝑉,π‘Ÿ,𝑦
Allowed substitution hints:   𝐴(𝑧,𝑀,𝑑,𝑠)   𝑅(𝑦,𝑧,𝑀,𝑑,𝑓,𝑠)   𝐹(π‘₯,𝑦,𝑧,𝑀,𝑑,𝑓,𝑠,π‘Ÿ)   𝑉(π‘₯,𝑧,𝑀,𝑑,𝑓,𝑠)
Proof of Theorem hartogslem1
StepHypRef Expression
1 hartogslem.2 . . . . 5 𝐹 = {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))}
21dmeqi 5903 . . . 4 dom 𝐹 = dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))}
3 dmopab 5914 . . . 4 dom {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} = {π‘Ÿ ∣ βˆƒπ‘¦(((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))}
42, 3eqtri 2761 . . 3 dom 𝐹 = {π‘Ÿ ∣ βˆƒπ‘¦(((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))}
5 simp3 1139 . . . . . . . 8 ((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) β†’ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ))
6 simp1 1137 . . . . . . . . 9 ((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) β†’ dom π‘Ÿ βŠ† 𝐴)
7 xpss12 5691 . . . . . . . . 9 ((dom π‘Ÿ βŠ† 𝐴 ∧ dom π‘Ÿ βŠ† 𝐴) β†’ (dom π‘Ÿ Γ— dom π‘Ÿ) βŠ† (𝐴 Γ— 𝐴))
86, 6, 7syl2anc 585 . . . . . . . 8 ((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) β†’ (dom π‘Ÿ Γ— dom π‘Ÿ) βŠ† (𝐴 Γ— 𝐴))
95, 8sstrd 3992 . . . . . . 7 ((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) β†’ π‘Ÿ βŠ† (𝐴 Γ— 𝐴))
10 velpw 4607 . . . . . . 7 (π‘Ÿ ∈ 𝒫 (𝐴 Γ— 𝐴) ↔ π‘Ÿ βŠ† (𝐴 Γ— 𝐴))
119, 10sylibr 233 . . . . . 6 ((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) β†’ π‘Ÿ ∈ 𝒫 (𝐴 Γ— 𝐴))
1211ad2antrr 725 . . . . 5 ((((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ)) β†’ π‘Ÿ ∈ 𝒫 (𝐴 Γ— 𝐴))
1312exlimiv 1934 . . . 4 (βˆƒπ‘¦(((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ)) β†’ π‘Ÿ ∈ 𝒫 (𝐴 Γ— 𝐴))
1413abssi 4067 . . 3 {π‘Ÿ ∣ βˆƒπ‘¦(((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} βŠ† 𝒫 (𝐴 Γ— 𝐴)
154, 14eqsstri 4016 . 2 dom 𝐹 βŠ† 𝒫 (𝐴 Γ— 𝐴)
16 funopab4 6583 . . 3 Fun {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))}
171funeqi 6567 . . 3 (Fun 𝐹 ↔ Fun {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))})
1816, 17mpbir 230 . 2 Fun 𝐹
191rneqi 5935 . . . 4 ran 𝐹 = ran {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))}
20 rnopab 5952 . . . 4 ran {βŸ¨π‘Ÿ, π‘¦βŸ© ∣ (((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))} = {𝑦 ∣ βˆƒπ‘Ÿ(((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))}
2119, 20eqtri 2761 . . 3 ran 𝐹 = {𝑦 ∣ βˆƒπ‘Ÿ(((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))}
22 breq1 5151 . . . . . 6 (π‘₯ = 𝑦 β†’ (π‘₯ β‰Ό 𝐴 ↔ 𝑦 β‰Ό 𝐴))
2322elrab 3683 . . . . 5 (𝑦 ∈ {π‘₯ ∈ On ∣ π‘₯ β‰Ό 𝐴} ↔ (𝑦 ∈ On ∧ 𝑦 β‰Ό 𝐴))
24 f1f 6785 . . . . . . . . . . . . 13 (𝑓:𝑦–1-1→𝐴 β†’ 𝑓:π‘¦βŸΆπ΄)
2524adantl 483 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ 𝑓:π‘¦βŸΆπ΄)
2625frnd 6723 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ ran 𝑓 βŠ† 𝐴)
27 resss 6005 . . . . . . . . . . . . . 14 ( I β†Ύ ran 𝑓) βŠ† I
28 ssun2 4173 . . . . . . . . . . . . . 14 I βŠ† (𝑅 βˆͺ I )
2927, 28sstri 3991 . . . . . . . . . . . . 13 ( I β†Ύ ran 𝑓) βŠ† (𝑅 βˆͺ I )
30 idssxp 6047 . . . . . . . . . . . . 13 ( I β†Ύ ran 𝑓) βŠ† (ran 𝑓 Γ— ran 𝑓)
3129, 30ssini 4231 . . . . . . . . . . . 12 ( I β†Ύ ran 𝑓) βŠ† ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓))
3231a1i 11 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ ( I β†Ύ ran 𝑓) βŠ† ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)))
33 inss2 4229 . . . . . . . . . . . 12 ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) βŠ† (ran 𝑓 Γ— ran 𝑓)
3433a1i 11 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) βŠ† (ran 𝑓 Γ— ran 𝑓))
3526, 32, 343jca 1129 . . . . . . . . . 10 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ (ran 𝑓 βŠ† 𝐴 ∧ ( I β†Ύ ran 𝑓) βŠ† ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) ∧ ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) βŠ† (ran 𝑓 Γ— ran 𝑓)))
36 eloni 6372 . . . . . . . . . . . . . 14 (𝑦 ∈ On β†’ Ord 𝑦)
37 ordwe 6375 . . . . . . . . . . . . . 14 (Ord 𝑦 β†’ E We 𝑦)
3836, 37syl 17 . . . . . . . . . . . . 13 (𝑦 ∈ On β†’ E We 𝑦)
3938adantr 482 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ E We 𝑦)
40 f1f1orn 6842 . . . . . . . . . . . . . . . 16 (𝑓:𝑦–1-1→𝐴 β†’ 𝑓:𝑦–1-1-ontoβ†’ran 𝑓)
4140adantl 483 . . . . . . . . . . . . . . 15 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ 𝑓:𝑦–1-1-ontoβ†’ran 𝑓)
42 hartogslem.3 . . . . . . . . . . . . . . 15 𝑅 = {βŸ¨π‘ , π‘‘βŸ© ∣ βˆƒπ‘€ ∈ 𝑦 βˆƒπ‘§ ∈ 𝑦 ((𝑠 = (π‘“β€˜π‘€) ∧ 𝑑 = (π‘“β€˜π‘§)) ∧ 𝑀 E 𝑧)}
43 f1oiso 7345 . . . . . . . . . . . . . . 15 ((𝑓:𝑦–1-1-ontoβ†’ran 𝑓 ∧ 𝑅 = {βŸ¨π‘ , π‘‘βŸ© ∣ βˆƒπ‘€ ∈ 𝑦 βˆƒπ‘§ ∈ 𝑦 ((𝑠 = (π‘“β€˜π‘€) ∧ 𝑑 = (π‘“β€˜π‘§)) ∧ 𝑀 E 𝑧)}) β†’ 𝑓 Isom E , 𝑅 (𝑦, ran 𝑓))
4441, 42, 43sylancl 587 . . . . . . . . . . . . . 14 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ 𝑓 Isom E , 𝑅 (𝑦, ran 𝑓))
45 isores2 7327 . . . . . . . . . . . . . 14 (𝑓 Isom E , 𝑅 (𝑦, ran 𝑓) ↔ 𝑓 Isom E , (𝑅 ∩ (ran 𝑓 Γ— ran 𝑓))(𝑦, ran 𝑓))
4644, 45sylib 217 . . . . . . . . . . . . 13 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ 𝑓 Isom E , (𝑅 ∩ (ran 𝑓 Γ— ran 𝑓))(𝑦, ran 𝑓))
47 isowe 7343 . . . . . . . . . . . . 13 (𝑓 Isom E , (𝑅 ∩ (ran 𝑓 Γ— ran 𝑓))(𝑦, ran 𝑓) β†’ ( E We 𝑦 ↔ (𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) We ran 𝑓))
4846, 47syl 17 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ ( E We 𝑦 ↔ (𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) We ran 𝑓))
4939, 48mpbid 231 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ (𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) We ran 𝑓)
50 weso 5667 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) We ran 𝑓 β†’ (𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) Or ran 𝑓)
5149, 50syl 17 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ (𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) Or ran 𝑓)
52 inss2 4229 . . . . . . . . . . . . . . . . . . 19 (𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βŠ† (ran 𝑓 Γ— ran 𝑓)
5352brel 5740 . . . . . . . . . . . . . . . . . 18 (π‘₯(𝑅 ∩ (ran 𝑓 Γ— ran 𝑓))π‘₯ β†’ (π‘₯ ∈ ran 𝑓 ∧ π‘₯ ∈ ran 𝑓))
5453simpld 496 . . . . . . . . . . . . . . . . 17 (π‘₯(𝑅 ∩ (ran 𝑓 Γ— ran 𝑓))π‘₯ β†’ π‘₯ ∈ ran 𝑓)
55 sonr 5611 . . . . . . . . . . . . . . . . 17 (((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) Or ran 𝑓 ∧ π‘₯ ∈ ran 𝑓) β†’ Β¬ π‘₯(𝑅 ∩ (ran 𝑓 Γ— ran 𝑓))π‘₯)
5651, 54, 55syl2an 597 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) ∧ π‘₯(𝑅 ∩ (ran 𝑓 Γ— ran 𝑓))π‘₯) β†’ Β¬ π‘₯(𝑅 ∩ (ran 𝑓 Γ— ran 𝑓))π‘₯)
5756pm2.01da 798 . . . . . . . . . . . . . . 15 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ Β¬ π‘₯(𝑅 ∩ (ran 𝑓 Γ— ran 𝑓))π‘₯)
5857alrimiv 1931 . . . . . . . . . . . . . 14 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ βˆ€π‘₯ Β¬ π‘₯(𝑅 ∩ (ran 𝑓 Γ— ran 𝑓))π‘₯)
59 intirr 6117 . . . . . . . . . . . . . 14 (((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) ∩ I ) = βˆ… ↔ βˆ€π‘₯ Β¬ π‘₯(𝑅 ∩ (ran 𝑓 Γ— ran 𝑓))π‘₯)
6058, 59sylibr 233 . . . . . . . . . . . . 13 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) ∩ I ) = βˆ…)
61 disj3 4453 . . . . . . . . . . . . 13 (((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) ∩ I ) = βˆ… ↔ (𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ))
6260, 61sylib 217 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ (𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ))
63 weeq1 5664 . . . . . . . . . . . 12 ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ) β†’ ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) We ran 𝑓 ↔ ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ) We ran 𝑓))
6462, 63syl 17 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) We ran 𝑓 ↔ ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ) We ran 𝑓))
6549, 64mpbid 231 . . . . . . . . . 10 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ) We ran 𝑓)
6636adantr 482 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ Ord 𝑦)
67 isoeq3 7313 . . . . . . . . . . . . . 14 ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ) β†’ (𝑓 Isom E , (𝑅 ∩ (ran 𝑓 Γ— ran 𝑓))(𝑦, ran 𝑓) ↔ 𝑓 Isom E , ( (𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I )(𝑦, ran 𝑓)))
6862, 67syl 17 . . . . . . . . . . . . 13 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ (𝑓 Isom E , (𝑅 ∩ (ran 𝑓 Γ— ran 𝑓))(𝑦, ran 𝑓) ↔ 𝑓 Isom E , ( (𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I )(𝑦, ran 𝑓)))
6946, 68mpbid 231 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ 𝑓 Isom E , ( (𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I )(𝑦, ran 𝑓))
70 vex 3479 . . . . . . . . . . . . . . 15 𝑓 ∈ V
7170rnex 7900 . . . . . . . . . . . . . 14 ran 𝑓 ∈ V
72 exse 5639 . . . . . . . . . . . . . 14 (ran 𝑓 ∈ V β†’ ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ) Se ran 𝑓)
7371, 72ax-mp 5 . . . . . . . . . . . . 13 ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ) Se ran 𝑓
74 eqid 2733 . . . . . . . . . . . . . 14 OrdIso(((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ), ran 𝑓) = OrdIso(((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ), ran 𝑓)
7574oieu 9531 . . . . . . . . . . . . 13 ((((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ) We ran 𝑓 ∧ ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ) Se ran 𝑓) β†’ ((Ord 𝑦 ∧ 𝑓 Isom E , ( (𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I )(𝑦, ran 𝑓)) ↔ (𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ), ran 𝑓) ∧ 𝑓 = OrdIso(((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ), ran 𝑓))))
7665, 73, 75sylancl 587 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ ((Ord 𝑦 ∧ 𝑓 Isom E , ( (𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I )(𝑦, ran 𝑓)) ↔ (𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ), ran 𝑓) ∧ 𝑓 = OrdIso(((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ), ran 𝑓))))
7766, 69, 76mpbi2and 711 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ (𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ), ran 𝑓) ∧ 𝑓 = OrdIso(((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ), ran 𝑓)))
7877simpld 496 . . . . . . . . . 10 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ 𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ), ran 𝑓))
7971, 71xpex 7737 . . . . . . . . . . . 12 (ran 𝑓 Γ— ran 𝑓) ∈ V
8079inex2 5318 . . . . . . . . . . 11 ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) ∈ V
81 sseq1 4007 . . . . . . . . . . . . . . . . . . 19 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ (π‘Ÿ βŠ† (ran 𝑓 Γ— ran 𝑓) ↔ ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) βŠ† (ran 𝑓 Γ— ran 𝑓)))
8233, 81mpbiri 258 . . . . . . . . . . . . . . . . . 18 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ π‘Ÿ βŠ† (ran 𝑓 Γ— ran 𝑓))
83 dmss 5901 . . . . . . . . . . . . . . . . . 18 (π‘Ÿ βŠ† (ran 𝑓 Γ— ran 𝑓) β†’ dom π‘Ÿ βŠ† dom (ran 𝑓 Γ— ran 𝑓))
8482, 83syl 17 . . . . . . . . . . . . . . . . 17 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ dom π‘Ÿ βŠ† dom (ran 𝑓 Γ— ran 𝑓))
85 dmxpid 5928 . . . . . . . . . . . . . . . . 17 dom (ran 𝑓 Γ— ran 𝑓) = ran 𝑓
8684, 85sseqtrdi 4032 . . . . . . . . . . . . . . . 16 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ dom π‘Ÿ βŠ† ran 𝑓)
87 dmresi 6050 . . . . . . . . . . . . . . . . 17 dom ( I β†Ύ ran 𝑓) = ran 𝑓
88 sseq2 4008 . . . . . . . . . . . . . . . . . . 19 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ (( I β†Ύ ran 𝑓) βŠ† π‘Ÿ ↔ ( I β†Ύ ran 𝑓) βŠ† ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓))))
8931, 88mpbiri 258 . . . . . . . . . . . . . . . . . 18 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ ( I β†Ύ ran 𝑓) βŠ† π‘Ÿ)
90 dmss 5901 . . . . . . . . . . . . . . . . . 18 (( I β†Ύ ran 𝑓) βŠ† π‘Ÿ β†’ dom ( I β†Ύ ran 𝑓) βŠ† dom π‘Ÿ)
9189, 90syl 17 . . . . . . . . . . . . . . . . 17 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ dom ( I β†Ύ ran 𝑓) βŠ† dom π‘Ÿ)
9287, 91eqsstrrid 4031 . . . . . . . . . . . . . . . 16 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ ran 𝑓 βŠ† dom π‘Ÿ)
9386, 92eqssd 3999 . . . . . . . . . . . . . . 15 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ dom π‘Ÿ = ran 𝑓)
9493sseq1d 4013 . . . . . . . . . . . . . 14 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ (dom π‘Ÿ βŠ† 𝐴 ↔ ran 𝑓 βŠ† 𝐴))
9593reseq2d 5980 . . . . . . . . . . . . . . 15 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ ( I β†Ύ dom π‘Ÿ) = ( I β†Ύ ran 𝑓))
96 id 22 . . . . . . . . . . . . . . 15 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)))
9795, 96sseq12d 4015 . . . . . . . . . . . . . 14 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ (( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ↔ ( I β†Ύ ran 𝑓) βŠ† ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓))))
9893sqxpeqd 5708 . . . . . . . . . . . . . . 15 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ (dom π‘Ÿ Γ— dom π‘Ÿ) = (ran 𝑓 Γ— ran 𝑓))
9996, 98sseq12d 4015 . . . . . . . . . . . . . 14 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ (π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ) ↔ ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) βŠ† (ran 𝑓 Γ— ran 𝑓)))
10094, 97, 993anbi123d 1437 . . . . . . . . . . . . 13 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ ((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ↔ (ran 𝑓 βŠ† 𝐴 ∧ ( I β†Ύ ran 𝑓) βŠ† ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) ∧ ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) βŠ† (ran 𝑓 Γ— ran 𝑓))))
101 difeq1 4115 . . . . . . . . . . . . . . . 16 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ (π‘Ÿ βˆ– I ) = (((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ))
102 difun2 4480 . . . . . . . . . . . . . . . . . 18 ((𝑅 βˆͺ I ) βˆ– I ) = (𝑅 βˆ– I )
103102ineq1i 4208 . . . . . . . . . . . . . . . . 17 (((𝑅 βˆͺ I ) βˆ– I ) ∩ (ran 𝑓 Γ— ran 𝑓)) = ((𝑅 βˆ– I ) ∩ (ran 𝑓 Γ— ran 𝑓))
104 indif1 4271 . . . . . . . . . . . . . . . . 17 (((𝑅 βˆͺ I ) βˆ– I ) ∩ (ran 𝑓 Γ— ran 𝑓)) = (((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I )
105 indif1 4271 . . . . . . . . . . . . . . . . 17 ((𝑅 βˆ– I ) ∩ (ran 𝑓 Γ— ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I )
106103, 104, 1053eqtr3i 2769 . . . . . . . . . . . . . . . 16 (((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ) = ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I )
107101, 106eqtrdi 2789 . . . . . . . . . . . . . . 15 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ (π‘Ÿ βˆ– I ) = ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ))
108 weeq1 5664 . . . . . . . . . . . . . . 15 ((π‘Ÿ βˆ– I ) = ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ) β†’ ((π‘Ÿ βˆ– I ) We dom π‘Ÿ ↔ ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ) We dom π‘Ÿ))
109107, 108syl 17 . . . . . . . . . . . . . 14 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ ((π‘Ÿ βˆ– I ) We dom π‘Ÿ ↔ ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ) We dom π‘Ÿ))
110 weeq2 5665 . . . . . . . . . . . . . . 15 (dom π‘Ÿ = ran 𝑓 β†’ (((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ) We dom π‘Ÿ ↔ ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ) We ran 𝑓))
11193, 110syl 17 . . . . . . . . . . . . . 14 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ (((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ) We dom π‘Ÿ ↔ ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ) We ran 𝑓))
112109, 111bitrd 279 . . . . . . . . . . . . 13 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ ((π‘Ÿ βˆ– I ) We dom π‘Ÿ ↔ ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ) We ran 𝑓))
113100, 112anbi12d 632 . . . . . . . . . . . 12 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ (((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ↔ ((ran 𝑓 βŠ† 𝐴 ∧ ( I β†Ύ ran 𝑓) βŠ† ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) ∧ ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) βŠ† (ran 𝑓 Γ— ran 𝑓)) ∧ ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ) We ran 𝑓)))
114 oieq1 9504 . . . . . . . . . . . . . . . 16 ((π‘Ÿ βˆ– I ) = ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ) β†’ OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ) = OrdIso(((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ), dom π‘Ÿ))
115107, 114syl 17 . . . . . . . . . . . . . . 15 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ) = OrdIso(((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ), dom π‘Ÿ))
116 oieq2 9505 . . . . . . . . . . . . . . . 16 (dom π‘Ÿ = ran 𝑓 β†’ OrdIso(((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ), dom π‘Ÿ) = OrdIso(((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ), ran 𝑓))
11793, 116syl 17 . . . . . . . . . . . . . . 15 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ OrdIso(((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ), dom π‘Ÿ) = OrdIso(((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ), ran 𝑓))
118115, 117eqtrd 2773 . . . . . . . . . . . . . 14 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ) = OrdIso(((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ), ran 𝑓))
119118dmeqd 5904 . . . . . . . . . . . . 13 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ) = dom OrdIso(((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ), ran 𝑓))
120119eqeq2d 2744 . . . . . . . . . . . 12 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ (𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ) ↔ 𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ), ran 𝑓)))
121113, 120anbi12d 632 . . . . . . . . . . 11 (π‘Ÿ = ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) β†’ ((((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ)) ↔ (((ran 𝑓 βŠ† 𝐴 ∧ ( I β†Ύ ran 𝑓) βŠ† ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) ∧ ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) βŠ† (ran 𝑓 Γ— ran 𝑓)) ∧ ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ) We ran 𝑓) ∧ 𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ), ran 𝑓))))
12280, 121spcev 3597 . . . . . . . . . 10 ((((ran 𝑓 βŠ† 𝐴 ∧ ( I β†Ύ ran 𝑓) βŠ† ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) ∧ ((𝑅 βˆͺ I ) ∩ (ran 𝑓 Γ— ran 𝑓)) βŠ† (ran 𝑓 Γ— ran 𝑓)) ∧ ((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ) We ran 𝑓) ∧ 𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 Γ— ran 𝑓)) βˆ– I ), ran 𝑓)) β†’ βˆƒπ‘Ÿ(((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ)))
12335, 65, 78, 122syl21anc 837 . . . . . . . . 9 ((𝑦 ∈ On ∧ 𝑓:𝑦–1-1→𝐴) β†’ βˆƒπ‘Ÿ(((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ)))
124123ex 414 . . . . . . . 8 (𝑦 ∈ On β†’ (𝑓:𝑦–1-1→𝐴 β†’ βˆƒπ‘Ÿ(((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))))
125124exlimdv 1937 . . . . . . 7 (𝑦 ∈ On β†’ (βˆƒπ‘“ 𝑓:𝑦–1-1→𝐴 β†’ βˆƒπ‘Ÿ(((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))))
126 brdomi 8951 . . . . . . 7 (𝑦 β‰Ό 𝐴 β†’ βˆƒπ‘“ 𝑓:𝑦–1-1→𝐴)
127125, 126impel 507 . . . . . 6 ((𝑦 ∈ On ∧ 𝑦 β‰Ό 𝐴) β†’ βˆƒπ‘Ÿ(((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ)))
128 simpr 486 . . . . . . . . . . 11 ((((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ)) β†’ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))
129 vex 3479 . . . . . . . . . . . . 13 π‘Ÿ ∈ V
130129dmex 7899 . . . . . . . . . . . 12 dom π‘Ÿ ∈ V
131 eqid 2733 . . . . . . . . . . . . 13 OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ) = OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ)
132131oion 9528 . . . . . . . . . . . 12 (dom π‘Ÿ ∈ V β†’ dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ) ∈ On)
133130, 132ax-mp 5 . . . . . . . . . . 11 dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ) ∈ On
134128, 133eqeltrdi 2842 . . . . . . . . . 10 ((((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ)) β†’ 𝑦 ∈ On)
135134adantl 483 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ (((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))) β†’ 𝑦 ∈ On)
136 simplr 768 . . . . . . . . . . . 12 ((((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ)) β†’ (π‘Ÿ βˆ– I ) We dom π‘Ÿ)
137131oien 9530 . . . . . . . . . . . 12 ((dom π‘Ÿ ∈ V ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) β†’ dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ) β‰ˆ dom π‘Ÿ)
138130, 136, 137sylancr 588 . . . . . . . . . . 11 ((((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ)) β†’ dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ) β‰ˆ dom π‘Ÿ)
139128, 138eqbrtrd 5170 . . . . . . . . . 10 ((((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ)) β†’ 𝑦 β‰ˆ dom π‘Ÿ)
140 ssdomg 8993 . . . . . . . . . . 11 (𝐴 ∈ 𝑉 β†’ (dom π‘Ÿ βŠ† 𝐴 β†’ dom π‘Ÿ β‰Ό 𝐴))
141 simpll1 1213 . . . . . . . . . . 11 ((((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ)) β†’ dom π‘Ÿ βŠ† 𝐴)
142140, 141impel 507 . . . . . . . . . 10 ((𝐴 ∈ 𝑉 ∧ (((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))) β†’ dom π‘Ÿ β‰Ό 𝐴)
143 endomtr 9005 . . . . . . . . . 10 ((𝑦 β‰ˆ dom π‘Ÿ ∧ dom π‘Ÿ β‰Ό 𝐴) β†’ 𝑦 β‰Ό 𝐴)
144139, 142, 143syl2an2 685 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ (((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))) β†’ 𝑦 β‰Ό 𝐴)
145135, 144jca 513 . . . . . . . 8 ((𝐴 ∈ 𝑉 ∧ (((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))) β†’ (𝑦 ∈ On ∧ 𝑦 β‰Ό 𝐴))
146145ex 414 . . . . . . 7 (𝐴 ∈ 𝑉 β†’ ((((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ)) β†’ (𝑦 ∈ On ∧ 𝑦 β‰Ό 𝐴)))
147146exlimdv 1937 . . . . . 6 (𝐴 ∈ 𝑉 β†’ (βˆƒπ‘Ÿ(((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ)) β†’ (𝑦 ∈ On ∧ 𝑦 β‰Ό 𝐴)))
148127, 147impbid2 225 . . . . 5 (𝐴 ∈ 𝑉 β†’ ((𝑦 ∈ On ∧ 𝑦 β‰Ό 𝐴) ↔ βˆƒπ‘Ÿ(((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))))
14923, 148bitrid 283 . . . 4 (𝐴 ∈ 𝑉 β†’ (𝑦 ∈ {π‘₯ ∈ On ∣ π‘₯ β‰Ό 𝐴} ↔ βˆƒπ‘Ÿ(((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))))
150149eqabdv 2868 . . 3 (𝐴 ∈ 𝑉 β†’ {π‘₯ ∈ On ∣ π‘₯ β‰Ό 𝐴} = {𝑦 ∣ βˆƒπ‘Ÿ(((dom π‘Ÿ βŠ† 𝐴 ∧ ( I β†Ύ dom π‘Ÿ) βŠ† π‘Ÿ ∧ π‘Ÿ βŠ† (dom π‘Ÿ Γ— dom π‘Ÿ)) ∧ (π‘Ÿ βˆ– I ) We dom π‘Ÿ) ∧ 𝑦 = dom OrdIso((π‘Ÿ βˆ– I ), dom π‘Ÿ))})
15121, 150eqtr4id 2792 . 2 (𝐴 ∈ 𝑉 β†’ ran 𝐹 = {π‘₯ ∈ On ∣ π‘₯ β‰Ό 𝐴})
15215, 18, 1513pm3.2i 1340 1 (dom 𝐹 βŠ† 𝒫 (𝐴 Γ— 𝐴) ∧ Fun 𝐹 ∧ (𝐴 ∈ 𝑉 β†’ ran 𝐹 = {π‘₯ ∈ On ∣ π‘₯ β‰Ό 𝐴}))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088  βˆ€wal 1540   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710  βˆƒwrex 3071  {crab 3433  Vcvv 3475   βˆ– cdif 3945   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602   class class class wbr 5148  {copab 5210   I cid 5573   E cep 5579   Or wor 5587   Se wse 5629   We wwe 5630   Γ— cxp 5674  dom cdm 5676  ran crn 5677   β†Ύ cres 5678  Ord word 6361  Oncon0 6362  Fun wfun 6535  βŸΆwf 6537  β€“1-1β†’wf1 6538  β€“1-1-ontoβ†’wf1o 6540  β€˜cfv 6541   Isom wiso 6542   β‰ˆ cen 8933   β‰Ό cdom 8934  OrdIsocoi 9501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7362  df-ov 7409  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-en 8937  df-dom 8938  df-oi 9502
This theorem is referenced by:  hartogslem2  9535  harwdom  9583
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