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Theorem hartogslem1 9039
 Description: Lemma for hartogs 9041. (Contributed by Mario Carneiro, 14-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
Hypotheses
Ref Expression
hartogslem.2 𝐹 = {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
hartogslem.3 𝑅 = {⟨𝑠, 𝑡⟩ ∣ ∃𝑤𝑦𝑧𝑦 ((𝑠 = (𝑓𝑤) ∧ 𝑡 = (𝑓𝑧)) ∧ 𝑤 E 𝑧)}
Assertion
Ref Expression
hartogslem1 (dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) ∧ Fun 𝐹 ∧ (𝐴𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥𝐴}))
Distinct variable groups:   𝑓,𝑠,𝑡,𝑤,𝑦,𝑧   𝑓,𝑟,𝑥,𝐴,𝑦   𝑅,𝑟,𝑥   𝑉,𝑟,𝑦
Allowed substitution hints:   𝐴(𝑧,𝑤,𝑡,𝑠)   𝑅(𝑦,𝑧,𝑤,𝑡,𝑓,𝑠)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑡,𝑓,𝑠,𝑟)   𝑉(𝑥,𝑧,𝑤,𝑡,𝑓,𝑠)

Proof of Theorem hartogslem1
StepHypRef Expression
1 hartogslem.2 . . . . 5 𝐹 = {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
21dmeqi 5744 . . . 4 dom 𝐹 = dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
3 dmopab 5755 . . . 4 dom {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} = {𝑟 ∣ ∃𝑦(((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
42, 3eqtri 2781 . . 3 dom 𝐹 = {𝑟 ∣ ∃𝑦(((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
5 simp3 1135 . . . . . . . 8 ((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → 𝑟 ⊆ (dom 𝑟 × dom 𝑟))
6 simp1 1133 . . . . . . . . 9 ((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → dom 𝑟𝐴)
7 xpss12 5539 . . . . . . . . 9 ((dom 𝑟𝐴 ∧ dom 𝑟𝐴) → (dom 𝑟 × dom 𝑟) ⊆ (𝐴 × 𝐴))
86, 6, 7syl2anc 587 . . . . . . . 8 ((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → (dom 𝑟 × dom 𝑟) ⊆ (𝐴 × 𝐴))
95, 8sstrd 3902 . . . . . . 7 ((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → 𝑟 ⊆ (𝐴 × 𝐴))
10 velpw 4499 . . . . . . 7 (𝑟 ∈ 𝒫 (𝐴 × 𝐴) ↔ 𝑟 ⊆ (𝐴 × 𝐴))
119, 10sylibr 237 . . . . . 6 ((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) → 𝑟 ∈ 𝒫 (𝐴 × 𝐴))
1211ad2antrr 725 . . . . 5 ((((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → 𝑟 ∈ 𝒫 (𝐴 × 𝐴))
1312exlimiv 1931 . . . 4 (∃𝑦(((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → 𝑟 ∈ 𝒫 (𝐴 × 𝐴))
1413abssi 3974 . . 3 {𝑟 ∣ ∃𝑦(((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} ⊆ 𝒫 (𝐴 × 𝐴)
154, 14eqsstri 3926 . 2 dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴)
16 funopab4 6372 . . 3 Fun {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
171funeqi 6356 . . 3 (Fun 𝐹 ↔ Fun {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))})
1816, 17mpbir 234 . 2 Fun 𝐹
191rneqi 5778 . . . 4 ran 𝐹 = ran {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
20 rnopab 5795 . . . 4 ran {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} = {𝑦 ∣ ∃𝑟(((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
2119, 20eqtri 2781 . . 3 ran 𝐹 = {𝑦 ∣ ∃𝑟(((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}
22 breq1 5035 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
2322elrab 3602 . . . . 5 (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ (𝑦 ∈ On ∧ 𝑦𝐴))
24 f1f 6560 . . . . . . . . . . . . 13 (𝑓:𝑦1-1𝐴𝑓:𝑦𝐴)
2524adantl 485 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → 𝑓:𝑦𝐴)
2625frnd 6505 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → ran 𝑓𝐴)
27 resss 5848 . . . . . . . . . . . . . 14 ( I ↾ ran 𝑓) ⊆ I
28 ssun2 4078 . . . . . . . . . . . . . 14 I ⊆ (𝑅 ∪ I )
2927, 28sstri 3901 . . . . . . . . . . . . 13 ( I ↾ ran 𝑓) ⊆ (𝑅 ∪ I )
30 idssxp 5888 . . . . . . . . . . . . 13 ( I ↾ ran 𝑓) ⊆ (ran 𝑓 × ran 𝑓)
3129, 30ssini 4136 . . . . . . . . . . . 12 ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓))
3231a1i 11 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)))
33 inss2 4134 . . . . . . . . . . . 12 ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)
3433a1i 11 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓))
3526, 32, 343jca 1125 . . . . . . . . . 10 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → (ran 𝑓𝐴 ∧ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)))
36 eloni 6179 . . . . . . . . . . . . . 14 (𝑦 ∈ On → Ord 𝑦)
37 ordwe 6182 . . . . . . . . . . . . . 14 (Ord 𝑦 → E We 𝑦)
3836, 37syl 17 . . . . . . . . . . . . 13 (𝑦 ∈ On → E We 𝑦)
3938adantr 484 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → E We 𝑦)
40 f1f1orn 6613 . . . . . . . . . . . . . . . 16 (𝑓:𝑦1-1𝐴𝑓:𝑦1-1-onto→ran 𝑓)
4140adantl 485 . . . . . . . . . . . . . . 15 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → 𝑓:𝑦1-1-onto→ran 𝑓)
42 hartogslem.3 . . . . . . . . . . . . . . 15 𝑅 = {⟨𝑠, 𝑡⟩ ∣ ∃𝑤𝑦𝑧𝑦 ((𝑠 = (𝑓𝑤) ∧ 𝑡 = (𝑓𝑧)) ∧ 𝑤 E 𝑧)}
43 f1oiso 7098 . . . . . . . . . . . . . . 15 ((𝑓:𝑦1-1-onto→ran 𝑓𝑅 = {⟨𝑠, 𝑡⟩ ∣ ∃𝑤𝑦𝑧𝑦 ((𝑠 = (𝑓𝑤) ∧ 𝑡 = (𝑓𝑧)) ∧ 𝑤 E 𝑧)}) → 𝑓 Isom E , 𝑅 (𝑦, ran 𝑓))
4441, 42, 43sylancl 589 . . . . . . . . . . . . . 14 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → 𝑓 Isom E , 𝑅 (𝑦, ran 𝑓))
45 isores2 7080 . . . . . . . . . . . . . 14 (𝑓 Isom E , 𝑅 (𝑦, ran 𝑓) ↔ 𝑓 Isom E , (𝑅 ∩ (ran 𝑓 × ran 𝑓))(𝑦, ran 𝑓))
4644, 45sylib 221 . . . . . . . . . . . . 13 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → 𝑓 Isom E , (𝑅 ∩ (ran 𝑓 × ran 𝑓))(𝑦, ran 𝑓))
47 isowe 7096 . . . . . . . . . . . . 13 (𝑓 Isom E , (𝑅 ∩ (ran 𝑓 × ran 𝑓))(𝑦, ran 𝑓) → ( E We 𝑦 ↔ (𝑅 ∩ (ran 𝑓 × ran 𝑓)) We ran 𝑓))
4846, 47syl 17 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → ( E We 𝑦 ↔ (𝑅 ∩ (ran 𝑓 × ran 𝑓)) We ran 𝑓))
4939, 48mpbid 235 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → (𝑅 ∩ (ran 𝑓 × ran 𝑓)) We ran 𝑓)
50 weso 5515 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) We ran 𝑓 → (𝑅 ∩ (ran 𝑓 × ran 𝑓)) Or ran 𝑓)
5149, 50syl 17 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → (𝑅 ∩ (ran 𝑓 × ran 𝑓)) Or ran 𝑓)
52 inss2 4134 . . . . . . . . . . . . . . . . . . 19 (𝑅 ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)
5352brel 5586 . . . . . . . . . . . . . . . . . 18 (𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥 → (𝑥 ∈ ran 𝑓𝑥 ∈ ran 𝑓))
5453simpld 498 . . . . . . . . . . . . . . . . 17 (𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥𝑥 ∈ ran 𝑓)
55 sonr 5465 . . . . . . . . . . . . . . . . 17 (((𝑅 ∩ (ran 𝑓 × ran 𝑓)) Or ran 𝑓𝑥 ∈ ran 𝑓) → ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥)
5651, 54, 55syl2an 598 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) ∧ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥) → ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥)
5756pm2.01da 798 . . . . . . . . . . . . . . 15 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥)
5857alrimiv 1928 . . . . . . . . . . . . . 14 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → ∀𝑥 ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥)
59 intirr 5950 . . . . . . . . . . . . . 14 (((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∩ I ) = ∅ ↔ ∀𝑥 ¬ 𝑥(𝑅 ∩ (ran 𝑓 × ran 𝑓))𝑥)
6058, 59sylibr 237 . . . . . . . . . . . . 13 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∩ I ) = ∅)
61 disj3 4350 . . . . . . . . . . . . 13 (((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∩ I ) = ∅ ↔ (𝑅 ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ))
6260, 61sylib 221 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → (𝑅 ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ))
63 weeq1 5512 . . . . . . . . . . . 12 ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) → ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) We ran 𝑓 ↔ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓))
6462, 63syl 17 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) We ran 𝑓 ↔ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓))
6549, 64mpbid 235 . . . . . . . . . 10 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓)
6636adantr 484 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → Ord 𝑦)
67 isoeq3 7066 . . . . . . . . . . . . . 14 ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) → (𝑓 Isom E , (𝑅 ∩ (ran 𝑓 × ran 𝑓))(𝑦, ran 𝑓) ↔ 𝑓 Isom E , ( (𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )(𝑦, ran 𝑓)))
6862, 67syl 17 . . . . . . . . . . . . 13 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → (𝑓 Isom E , (𝑅 ∩ (ran 𝑓 × ran 𝑓))(𝑦, ran 𝑓) ↔ 𝑓 Isom E , ( (𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )(𝑦, ran 𝑓)))
6946, 68mpbid 235 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → 𝑓 Isom E , ( (𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )(𝑦, ran 𝑓))
70 vex 3413 . . . . . . . . . . . . . . 15 𝑓 ∈ V
7170rnex 7622 . . . . . . . . . . . . . 14 ran 𝑓 ∈ V
72 exse 5488 . . . . . . . . . . . . . 14 (ran 𝑓 ∈ V → ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) Se ran 𝑓)
7371, 72ax-mp 5 . . . . . . . . . . . . 13 ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) Se ran 𝑓
74 eqid 2758 . . . . . . . . . . . . . 14 OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓) = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓)
7574oieu 9036 . . . . . . . . . . . . 13 ((((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓 ∧ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) Se ran 𝑓) → ((Ord 𝑦𝑓 Isom E , ( (𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )(𝑦, ran 𝑓)) ↔ (𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓) ∧ 𝑓 = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓))))
7665, 73, 75sylancl 589 . . . . . . . . . . . 12 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → ((Ord 𝑦𝑓 Isom E , ( (𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )(𝑦, ran 𝑓)) ↔ (𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓) ∧ 𝑓 = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓))))
7766, 69, 76mpbi2and 711 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → (𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓) ∧ 𝑓 = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓)))
7877simpld 498 . . . . . . . . . 10 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → 𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓))
7971, 71xpex 7474 . . . . . . . . . . . 12 (ran 𝑓 × ran 𝑓) ∈ V
8079inex2 5188 . . . . . . . . . . 11 ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∈ V
81 sseq1 3917 . . . . . . . . . . . . . . . . . . 19 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (𝑟 ⊆ (ran 𝑓 × ran 𝑓) ↔ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)))
8233, 81mpbiri 261 . . . . . . . . . . . . . . . . . 18 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → 𝑟 ⊆ (ran 𝑓 × ran 𝑓))
83 dmss 5742 . . . . . . . . . . . . . . . . . 18 (𝑟 ⊆ (ran 𝑓 × ran 𝑓) → dom 𝑟 ⊆ dom (ran 𝑓 × ran 𝑓))
8482, 83syl 17 . . . . . . . . . . . . . . . . 17 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → dom 𝑟 ⊆ dom (ran 𝑓 × ran 𝑓))
85 dmxpid 5771 . . . . . . . . . . . . . . . . 17 dom (ran 𝑓 × ran 𝑓) = ran 𝑓
8684, 85sseqtrdi 3942 . . . . . . . . . . . . . . . 16 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → dom 𝑟 ⊆ ran 𝑓)
87 dmresi 5893 . . . . . . . . . . . . . . . . 17 dom ( I ↾ ran 𝑓) = ran 𝑓
88 sseq2 3918 . . . . . . . . . . . . . . . . . . 19 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (( I ↾ ran 𝑓) ⊆ 𝑟 ↔ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓))))
8931, 88mpbiri 261 . . . . . . . . . . . . . . . . . 18 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ( I ↾ ran 𝑓) ⊆ 𝑟)
90 dmss 5742 . . . . . . . . . . . . . . . . . 18 (( I ↾ ran 𝑓) ⊆ 𝑟 → dom ( I ↾ ran 𝑓) ⊆ dom 𝑟)
9189, 90syl 17 . . . . . . . . . . . . . . . . 17 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → dom ( I ↾ ran 𝑓) ⊆ dom 𝑟)
9287, 91eqsstrrid 3941 . . . . . . . . . . . . . . . 16 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ran 𝑓 ⊆ dom 𝑟)
9386, 92eqssd 3909 . . . . . . . . . . . . . . 15 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → dom 𝑟 = ran 𝑓)
9493sseq1d 3923 . . . . . . . . . . . . . 14 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (dom 𝑟𝐴 ↔ ran 𝑓𝐴))
9593reseq2d 5823 . . . . . . . . . . . . . . 15 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ( I ↾ dom 𝑟) = ( I ↾ ran 𝑓))
96 id 22 . . . . . . . . . . . . . . 15 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → 𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)))
9795, 96sseq12d 3925 . . . . . . . . . . . . . 14 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓))))
9893sqxpeqd 5556 . . . . . . . . . . . . . . 15 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (dom 𝑟 × dom 𝑟) = (ran 𝑓 × ran 𝑓))
9996, 98sseq12d 3925 . . . . . . . . . . . . . 14 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (𝑟 ⊆ (dom 𝑟 × dom 𝑟) ↔ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)))
10094, 97, 993anbi123d 1433 . . . . . . . . . . . . 13 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ↔ (ran 𝑓𝐴 ∧ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓))))
101 difeq1 4021 . . . . . . . . . . . . . . . 16 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (𝑟 ∖ I ) = (((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∖ I ))
102 difun2 4377 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∪ I ) ∖ I ) = (𝑅 ∖ I )
103102ineq1i 4113 . . . . . . . . . . . . . . . . 17 (((𝑅 ∪ I ) ∖ I ) ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∖ I ) ∩ (ran 𝑓 × ran 𝑓))
104 indif1 4176 . . . . . . . . . . . . . . . . 17 (((𝑅 ∪ I ) ∖ I ) ∩ (ran 𝑓 × ran 𝑓)) = (((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∖ I )
105 indif1 4176 . . . . . . . . . . . . . . . . 17 ((𝑅 ∖ I ) ∩ (ran 𝑓 × ran 𝑓)) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )
106103, 104, 1053eqtr3i 2789 . . . . . . . . . . . . . . . 16 (((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I )
107101, 106eqtrdi 2809 . . . . . . . . . . . . . . 15 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (𝑟 ∖ I ) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ))
108 weeq1 5512 . . . . . . . . . . . . . . 15 ((𝑟 ∖ I ) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) → ((𝑟 ∖ I ) We dom 𝑟 ↔ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We dom 𝑟))
109107, 108syl 17 . . . . . . . . . . . . . 14 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ((𝑟 ∖ I ) We dom 𝑟 ↔ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We dom 𝑟))
110 weeq2 5513 . . . . . . . . . . . . . . 15 (dom 𝑟 = ran 𝑓 → (((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We dom 𝑟 ↔ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓))
11193, 110syl 17 . . . . . . . . . . . . . 14 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We dom 𝑟 ↔ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓))
112109, 111bitrd 282 . . . . . . . . . . . . 13 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ((𝑟 ∖ I ) We dom 𝑟 ↔ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓))
113100, 112anbi12d 633 . . . . . . . . . . . 12 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ↔ ((ran 𝑓𝐴 ∧ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓)))
114 oieq1 9009 . . . . . . . . . . . . . . . 16 ((𝑟 ∖ I ) = ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) → OrdIso((𝑟 ∖ I ), dom 𝑟) = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), dom 𝑟))
115107, 114syl 17 . . . . . . . . . . . . . . 15 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → OrdIso((𝑟 ∖ I ), dom 𝑟) = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), dom 𝑟))
116 oieq2 9010 . . . . . . . . . . . . . . . 16 (dom 𝑟 = ran 𝑓 → OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), dom 𝑟) = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓))
11793, 116syl 17 . . . . . . . . . . . . . . 15 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), dom 𝑟) = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓))
118115, 117eqtrd 2793 . . . . . . . . . . . . . 14 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → OrdIso((𝑟 ∖ I ), dom 𝑟) = OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓))
119118dmeqd 5745 . . . . . . . . . . . . 13 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → dom OrdIso((𝑟 ∖ I ), dom 𝑟) = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓))
120119eqeq2d 2769 . . . . . . . . . . . 12 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → (𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟) ↔ 𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓)))
121113, 120anbi12d 633 . . . . . . . . . . 11 (𝑟 = ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) → ((((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) ↔ (((ran 𝑓𝐴 ∧ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓) ∧ 𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓))))
12280, 121spcev 3525 . . . . . . . . . 10 ((((ran 𝑓𝐴 ∧ ( I ↾ ran 𝑓) ⊆ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∪ I ) ∩ (ran 𝑓 × ran 𝑓)) ⊆ (ran 𝑓 × ran 𝑓)) ∧ ((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ) We ran 𝑓) ∧ 𝑦 = dom OrdIso(((𝑅 ∩ (ran 𝑓 × ran 𝑓)) ∖ I ), ran 𝑓)) → ∃𝑟(((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)))
12335, 65, 78, 122syl21anc 836 . . . . . . . . 9 ((𝑦 ∈ On ∧ 𝑓:𝑦1-1𝐴) → ∃𝑟(((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)))
124123ex 416 . . . . . . . 8 (𝑦 ∈ On → (𝑓:𝑦1-1𝐴 → ∃𝑟(((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))))
125124exlimdv 1934 . . . . . . 7 (𝑦 ∈ On → (∃𝑓 𝑓:𝑦1-1𝐴 → ∃𝑟(((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))))
126 brdomi 8538 . . . . . . 7 (𝑦𝐴 → ∃𝑓 𝑓:𝑦1-1𝐴)
127125, 126impel 509 . . . . . 6 ((𝑦 ∈ On ∧ 𝑦𝐴) → ∃𝑟(((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)))
128 simpr 488 . . . . . . . . . . 11 ((((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))
129 vex 3413 . . . . . . . . . . . . 13 𝑟 ∈ V
130129dmex 7621 . . . . . . . . . . . 12 dom 𝑟 ∈ V
131 eqid 2758 . . . . . . . . . . . . 13 OrdIso((𝑟 ∖ I ), dom 𝑟) = OrdIso((𝑟 ∖ I ), dom 𝑟)
132131oion 9033 . . . . . . . . . . . 12 (dom 𝑟 ∈ V → dom OrdIso((𝑟 ∖ I ), dom 𝑟) ∈ On)
133130, 132ax-mp 5 . . . . . . . . . . 11 dom OrdIso((𝑟 ∖ I ), dom 𝑟) ∈ On
134128, 133eqeltrdi 2860 . . . . . . . . . 10 ((((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → 𝑦 ∈ On)
135134adantl 485 . . . . . . . . 9 ((𝐴𝑉 ∧ (((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))) → 𝑦 ∈ On)
136 simplr 768 . . . . . . . . . . . 12 ((((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → (𝑟 ∖ I ) We dom 𝑟)
137131oien 9035 . . . . . . . . . . . 12 ((dom 𝑟 ∈ V ∧ (𝑟 ∖ I ) We dom 𝑟) → dom OrdIso((𝑟 ∖ I ), dom 𝑟) ≈ dom 𝑟)
138130, 136, 137sylancr 590 . . . . . . . . . . 11 ((((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → dom OrdIso((𝑟 ∖ I ), dom 𝑟) ≈ dom 𝑟)
139128, 138eqbrtrd 5054 . . . . . . . . . 10 ((((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → 𝑦 ≈ dom 𝑟)
140 ssdomg 8573 . . . . . . . . . . 11 (𝐴𝑉 → (dom 𝑟𝐴 → dom 𝑟𝐴))
141 simpll1 1209 . . . . . . . . . . 11 ((((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → dom 𝑟𝐴)
142140, 141impel 509 . . . . . . . . . 10 ((𝐴𝑉 ∧ (((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))) → dom 𝑟𝐴)
143 endomtr 8585 . . . . . . . . . 10 ((𝑦 ≈ dom 𝑟 ∧ dom 𝑟𝐴) → 𝑦𝐴)
144139, 142, 143syl2an2 685 . . . . . . . . 9 ((𝐴𝑉 ∧ (((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))) → 𝑦𝐴)
145135, 144jca 515 . . . . . . . 8 ((𝐴𝑉 ∧ (((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))) → (𝑦 ∈ On ∧ 𝑦𝐴))
146145ex 416 . . . . . . 7 (𝐴𝑉 → ((((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → (𝑦 ∈ On ∧ 𝑦𝐴)))
147146exlimdv 1934 . . . . . 6 (𝐴𝑉 → (∃𝑟(((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟)) → (𝑦 ∈ On ∧ 𝑦𝐴)))
148127, 147impbid2 229 . . . . 5 (𝐴𝑉 → ((𝑦 ∈ On ∧ 𝑦𝐴) ↔ ∃𝑟(((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))))
14923, 148syl5bb 286 . . . 4 (𝐴𝑉 → (𝑦 ∈ {𝑥 ∈ On ∣ 𝑥𝐴} ↔ ∃𝑟(((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))))
150149abbi2dv 2889 . . 3 (𝐴𝑉 → {𝑥 ∈ On ∣ 𝑥𝐴} = {𝑦 ∣ ∃𝑟(((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))})
15121, 150eqtr4id 2812 . 2 (𝐴𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥𝐴})
15215, 18, 1513pm3.2i 1336 1 (dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) ∧ Fun 𝐹 ∧ (𝐴𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥𝐴}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2735  ∃wrex 3071  {crab 3074  Vcvv 3409   ∖ cdif 3855   ∪ cun 3856   ∩ cin 3857   ⊆ wss 3858  ∅c0 4225  𝒫 cpw 4494   class class class wbr 5032  {copab 5094   I cid 5429   E cep 5434   Or wor 5442   Se wse 5481   We wwe 5482   × cxp 5522  dom cdm 5524  ran crn 5525   ↾ cres 5526  Ord word 6168  Oncon0 6169  Fun wfun 6329  ⟶wf 6331  –1-1→wf1 6332  –1-1-onto→wf1o 6334  ‘cfv 6335   Isom wiso 6336   ≈ cen 8524   ≼ cdom 8525  OrdIsocoi 9006 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-se 5484  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-isom 6344  df-riota 7108  df-wrecs 7957  df-recs 8018  df-en 8528  df-dom 8529  df-oi 9007 This theorem is referenced by:  hartogslem2  9040  harwdom  9088
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