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Theorem rankonidlem 9868
Description: Lemma for rankonid 9869. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.)
Assertion
Ref Expression
rankonidlem (𝐴 ∈ dom 𝑅1 → (𝐴 (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))

Proof of Theorem rankonidlem
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r1funlim 9806 . . . . 5 (Fun 𝑅1 ∧ Lim dom 𝑅1)
21simpri 485 . . . 4 Lim dom 𝑅1
3 limord 6444 . . . 4 (Lim dom 𝑅1 → Ord dom 𝑅1)
42, 3ax-mp 5 . . 3 Ord dom 𝑅1
5 ordelon 6408 . . 3 ((Ord dom 𝑅1𝐴 ∈ dom 𝑅1) → 𝐴 ∈ On)
64, 5mpan 690 . 2 (𝐴 ∈ dom 𝑅1𝐴 ∈ On)
7 eleq1 2829 . . . 4 (𝑥 = 𝑦 → (𝑥 ∈ dom 𝑅1𝑦 ∈ dom 𝑅1))
8 eleq1 2829 . . . . 5 (𝑥 = 𝑦 → (𝑥 (𝑅1 “ On) ↔ 𝑦 (𝑅1 “ On)))
9 fveq2 6906 . . . . . 6 (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦))
10 id 22 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
119, 10eqeq12d 2753 . . . . 5 (𝑥 = 𝑦 → ((rank‘𝑥) = 𝑥 ↔ (rank‘𝑦) = 𝑦))
128, 11anbi12d 632 . . . 4 (𝑥 = 𝑦 → ((𝑥 (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥) ↔ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)))
137, 12imbi12d 344 . . 3 (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝑅1 → (𝑥 (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)) ↔ (𝑦 ∈ dom 𝑅1 → (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦))))
14 eleq1 2829 . . . 4 (𝑥 = 𝐴 → (𝑥 ∈ dom 𝑅1𝐴 ∈ dom 𝑅1))
15 eleq1 2829 . . . . 5 (𝑥 = 𝐴 → (𝑥 (𝑅1 “ On) ↔ 𝐴 (𝑅1 “ On)))
16 fveq2 6906 . . . . . 6 (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
17 id 22 . . . . . 6 (𝑥 = 𝐴𝑥 = 𝐴)
1816, 17eqeq12d 2753 . . . . 5 (𝑥 = 𝐴 → ((rank‘𝑥) = 𝑥 ↔ (rank‘𝐴) = 𝐴))
1915, 18anbi12d 632 . . . 4 (𝑥 = 𝐴 → ((𝑥 (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥) ↔ (𝐴 (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴)))
2014, 19imbi12d 344 . . 3 (𝑥 = 𝐴 → ((𝑥 ∈ dom 𝑅1 → (𝑥 (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)) ↔ (𝐴 ∈ dom 𝑅1 → (𝐴 (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))))
21 ordtr1 6427 . . . . . . . . . 10 (Ord dom 𝑅1 → ((𝑦𝑥𝑥 ∈ dom 𝑅1) → 𝑦 ∈ dom 𝑅1))
224, 21ax-mp 5 . . . . . . . . 9 ((𝑦𝑥𝑥 ∈ dom 𝑅1) → 𝑦 ∈ dom 𝑅1)
2322ancoms 458 . . . . . . . 8 ((𝑥 ∈ dom 𝑅1𝑦𝑥) → 𝑦 ∈ dom 𝑅1)
24 pm5.5 361 . . . . . . . 8 (𝑦 ∈ dom 𝑅1 → ((𝑦 ∈ dom 𝑅1 → (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) ↔ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)))
2523, 24syl 17 . . . . . . 7 ((𝑥 ∈ dom 𝑅1𝑦𝑥) → ((𝑦 ∈ dom 𝑅1 → (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) ↔ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)))
2625ralbidva 3176 . . . . . 6 (𝑥 ∈ dom 𝑅1 → (∀𝑦𝑥 (𝑦 ∈ dom 𝑅1 → (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) ↔ ∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)))
27 simplr 769 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦𝑥)
28 ordelon 6408 . . . . . . . . . . . . . . . . . . . . . 22 ((Ord dom 𝑅1𝑥 ∈ dom 𝑅1) → 𝑥 ∈ On)
294, 28mpan 690 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ dom 𝑅1𝑥 ∈ On)
3029ad2antrr 726 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ On)
31 eloni 6394 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ On → Ord 𝑥)
3230, 31syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → Ord 𝑥)
33 ordelsuc 7840 . . . . . . . . . . . . . . . . . . 19 ((𝑦𝑥 ∧ Ord 𝑥) → (𝑦𝑥 ↔ suc 𝑦𝑥))
3427, 32, 33syl2anc 584 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑦𝑥 ↔ suc 𝑦𝑥))
3527, 34mpbid 232 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc 𝑦𝑥)
3623adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ dom 𝑅1)
37 limsuc 7870 . . . . . . . . . . . . . . . . . . . 20 (Lim dom 𝑅1 → (𝑦 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1))
382, 37ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1)
3936, 38sylib 218 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc 𝑦 ∈ dom 𝑅1)
40 simpll 767 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ dom 𝑅1)
41 r1ord3g 9819 . . . . . . . . . . . . . . . . . 18 ((suc 𝑦 ∈ dom 𝑅1𝑥 ∈ dom 𝑅1) → (suc 𝑦𝑥 → (𝑅1‘suc 𝑦) ⊆ (𝑅1𝑥)))
4239, 40, 41syl2anc 584 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (suc 𝑦𝑥 → (𝑅1‘suc 𝑦) ⊆ (𝑅1𝑥)))
4335, 42mpd 15 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅1‘suc 𝑦) ⊆ (𝑅1𝑥))
44 rankidb 9840 . . . . . . . . . . . . . . . . . 18 (𝑦 (𝑅1 “ On) → 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)))
4544ad2antrl 728 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)))
46 suceq 6450 . . . . . . . . . . . . . . . . . . 19 ((rank‘𝑦) = 𝑦 → suc (rank‘𝑦) = suc 𝑦)
4746ad2antll 729 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc (rank‘𝑦) = suc 𝑦)
4847fveq2d 6910 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅1‘suc (rank‘𝑦)) = (𝑅1‘suc 𝑦))
4945, 48eleqtrd 2843 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘suc 𝑦))
5043, 49sseldd 3984 . . . . . . . . . . . . . . 15 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1𝑥))
5150ex 412 . . . . . . . . . . . . . 14 ((𝑥 ∈ dom 𝑅1𝑦𝑥) → ((𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → 𝑦 ∈ (𝑅1𝑥)))
5251ralimdva 3167 . . . . . . . . . . . . 13 (𝑥 ∈ dom 𝑅1 → (∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ∀𝑦𝑥 𝑦 ∈ (𝑅1𝑥)))
5352imp 406 . . . . . . . . . . . 12 ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → ∀𝑦𝑥 𝑦 ∈ (𝑅1𝑥))
54 dfss3 3972 . . . . . . . . . . . 12 (𝑥 ⊆ (𝑅1𝑥) ↔ ∀𝑦𝑥 𝑦 ∈ (𝑅1𝑥))
5553, 54sylibr 234 . . . . . . . . . . 11 ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ⊆ (𝑅1𝑥))
56 vex 3484 . . . . . . . . . . . 12 𝑥 ∈ V
5756elpw 4604 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 (𝑅1𝑥) ↔ 𝑥 ⊆ (𝑅1𝑥))
5855, 57sylibr 234 . . . . . . . . . 10 ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ 𝒫 (𝑅1𝑥))
59 r1sucg 9809 . . . . . . . . . . 11 (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
6059adantr 480 . . . . . . . . . 10 ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
6158, 60eleqtrrd 2844 . . . . . . . . 9 ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ (𝑅1‘suc 𝑥))
62 r1elwf 9836 . . . . . . . . 9 (𝑥 ∈ (𝑅1‘suc 𝑥) → 𝑥 (𝑅1 “ On))
6361, 62syl 17 . . . . . . . 8 ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 (𝑅1 “ On))
64 rankval3b 9866 . . . . . . . . . 10 (𝑥 (𝑅1 “ On) → (rank‘𝑥) = {𝑧 ∈ On ∣ ∀𝑦𝑥 (rank‘𝑦) ∈ 𝑧})
6563, 64syl 17 . . . . . . . . 9 ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (rank‘𝑥) = {𝑧 ∈ On ∣ ∀𝑦𝑥 (rank‘𝑦) ∈ 𝑧})
66 eleq1 2829 . . . . . . . . . . . . . . . 16 ((rank‘𝑦) = 𝑦 → ((rank‘𝑦) ∈ 𝑧𝑦𝑧))
6766adantl 481 . . . . . . . . . . . . . . 15 ((𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ((rank‘𝑦) ∈ 𝑧𝑦𝑧))
6867ralimi 3083 . . . . . . . . . . . . . 14 (∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ∀𝑦𝑥 ((rank‘𝑦) ∈ 𝑧𝑦𝑧))
69 ralbi 3103 . . . . . . . . . . . . . 14 (∀𝑦𝑥 ((rank‘𝑦) ∈ 𝑧𝑦𝑧) → (∀𝑦𝑥 (rank‘𝑦) ∈ 𝑧 ↔ ∀𝑦𝑥 𝑦𝑧))
7068, 69syl 17 . . . . . . . . . . . . 13 (∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → (∀𝑦𝑥 (rank‘𝑦) ∈ 𝑧 ↔ ∀𝑦𝑥 𝑦𝑧))
71 dfss3 3972 . . . . . . . . . . . . 13 (𝑥𝑧 ↔ ∀𝑦𝑥 𝑦𝑧)
7270, 71bitr4di 289 . . . . . . . . . . . 12 (∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → (∀𝑦𝑥 (rank‘𝑦) ∈ 𝑧𝑥𝑧))
7372rabbidv 3444 . . . . . . . . . . 11 (∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → {𝑧 ∈ On ∣ ∀𝑦𝑥 (rank‘𝑦) ∈ 𝑧} = {𝑧 ∈ On ∣ 𝑥𝑧})
7473inteqd 4951 . . . . . . . . . 10 (∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → {𝑧 ∈ On ∣ ∀𝑦𝑥 (rank‘𝑦) ∈ 𝑧} = {𝑧 ∈ On ∣ 𝑥𝑧})
7574adantl 481 . . . . . . . . 9 ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → {𝑧 ∈ On ∣ ∀𝑦𝑥 (rank‘𝑦) ∈ 𝑧} = {𝑧 ∈ On ∣ 𝑥𝑧})
7629adantr 480 . . . . . . . . . 10 ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ On)
77 intmin 4968 . . . . . . . . . 10 (𝑥 ∈ On → {𝑧 ∈ On ∣ 𝑥𝑧} = 𝑥)
7876, 77syl 17 . . . . . . . . 9 ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → {𝑧 ∈ On ∣ 𝑥𝑧} = 𝑥)
7965, 75, 783eqtrd 2781 . . . . . . . 8 ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (rank‘𝑥) = 𝑥)
8063, 79jca 511 . . . . . . 7 ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑥 (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥))
8180ex 412 . . . . . 6 (𝑥 ∈ dom 𝑅1 → (∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → (𝑥 (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)))
8226, 81sylbid 240 . . . . 5 (𝑥 ∈ dom 𝑅1 → (∀𝑦𝑥 (𝑦 ∈ dom 𝑅1 → (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑥 (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)))
8382com12 32 . . . 4 (∀𝑦𝑥 (𝑦 ∈ dom 𝑅1 → (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑥 ∈ dom 𝑅1 → (𝑥 (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)))
8483a1i 11 . . 3 (𝑥 ∈ On → (∀𝑦𝑥 (𝑦 ∈ dom 𝑅1 → (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑥 ∈ dom 𝑅1 → (𝑥 (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥))))
8513, 20, 84tfis3 7879 . 2 (𝐴 ∈ On → (𝐴 ∈ dom 𝑅1 → (𝐴 (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴)))
866, 85mpcom 38 1 (𝐴 ∈ dom 𝑅1 → (𝐴 (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436  wss 3951  𝒫 cpw 4600   cuni 4907   cint 4946  dom cdm 5685  cima 5688  Ord word 6383  Oncon0 6384  Lim wlim 6385  suc csuc 6386  Fun wfun 6555  cfv 6561  𝑅1cr1 9802  rankcrnk 9803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-r1 9804  df-rank 9805
This theorem is referenced by:  rankonid  9869  onwf  9870  onssr1  9871
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