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Theorem rankonidlem 8574
 Description: Lemma for rankonid 8575. (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 8512 . . . . 5 (Fun 𝑅1 ∧ Lim dom 𝑅1)
21simpri 477 . . . 4 Lim dom 𝑅1
3 limord 5701 . . . 4 (Lim dom 𝑅1 → Ord dom 𝑅1)
42, 3ax-mp 5 . . 3 Ord dom 𝑅1
5 ordelon 5664 . . 3 ((Ord dom 𝑅1𝐴 ∈ dom 𝑅1) → 𝐴 ∈ On)
64, 5mpan 702 . 2 (𝐴 ∈ dom 𝑅1𝐴 ∈ On)
7 eleq1 2676 . . . 4 (𝑥 = 𝑦 → (𝑥 ∈ dom 𝑅1𝑦 ∈ dom 𝑅1))
8 eleq1 2676 . . . . 5 (𝑥 = 𝑦 → (𝑥 (𝑅1 “ On) ↔ 𝑦 (𝑅1 “ On)))
9 fveq2 6103 . . . . . 6 (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦))
10 id 22 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
119, 10eqeq12d 2625 . . . . 5 (𝑥 = 𝑦 → ((rank‘𝑥) = 𝑥 ↔ (rank‘𝑦) = 𝑦))
128, 11anbi12d 743 . . . 4 (𝑥 = 𝑦 → ((𝑥 (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥) ↔ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)))
137, 12imbi12d 333 . . 3 (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝑅1 → (𝑥 (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)) ↔ (𝑦 ∈ dom 𝑅1 → (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦))))
14 eleq1 2676 . . . 4 (𝑥 = 𝐴 → (𝑥 ∈ dom 𝑅1𝐴 ∈ dom 𝑅1))
15 eleq1 2676 . . . . 5 (𝑥 = 𝐴 → (𝑥 (𝑅1 “ On) ↔ 𝐴 (𝑅1 “ On)))
16 fveq2 6103 . . . . . 6 (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
17 id 22 . . . . . 6 (𝑥 = 𝐴𝑥 = 𝐴)
1816, 17eqeq12d 2625 . . . . 5 (𝑥 = 𝐴 → ((rank‘𝑥) = 𝑥 ↔ (rank‘𝐴) = 𝐴))
1915, 18anbi12d 743 . . . 4 (𝑥 = 𝐴 → ((𝑥 (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥) ↔ (𝐴 (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴)))
2014, 19imbi12d 333 . . 3 (𝑥 = 𝐴 → ((𝑥 ∈ dom 𝑅1 → (𝑥 (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)) ↔ (𝐴 ∈ dom 𝑅1 → (𝐴 (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))))
21 ordtr1 5684 . . . . . . . . . 10 (Ord dom 𝑅1 → ((𝑦𝑥𝑥 ∈ dom 𝑅1) → 𝑦 ∈ dom 𝑅1))
224, 21ax-mp 5 . . . . . . . . 9 ((𝑦𝑥𝑥 ∈ dom 𝑅1) → 𝑦 ∈ dom 𝑅1)
2322ancoms 468 . . . . . . . 8 ((𝑥 ∈ dom 𝑅1𝑦𝑥) → 𝑦 ∈ dom 𝑅1)
24 pm5.5 350 . . . . . . . 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 2968 . . . . . 6 (𝑥 ∈ dom 𝑅1 → (∀𝑦𝑥 (𝑦 ∈ dom 𝑅1 → (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) ↔ ∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)))
27 simplr 788 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦𝑥)
28 ordelon 5664 . . . . . . . . . . . . . . . . . . . . . 22 ((Ord dom 𝑅1𝑥 ∈ dom 𝑅1) → 𝑥 ∈ On)
294, 28mpan 702 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ dom 𝑅1𝑥 ∈ On)
3029ad2antrr 758 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ On)
31 eloni 5650 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ On → Ord 𝑥)
3230, 31syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → Ord 𝑥)
33 ordelsuc 6912 . . . . . . . . . . . . . . . . . . 19 ((𝑦𝑥 ∧ Ord 𝑥) → (𝑦𝑥 ↔ suc 𝑦𝑥))
3427, 32, 33syl2anc 691 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑦𝑥 ↔ suc 𝑦𝑥))
3527, 34mpbid 221 . . . . . . . . . . . . . . . . 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 6941 . . . . . . . . . . . . . . . . . . . 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 207 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc 𝑦 ∈ dom 𝑅1)
40 simpll 786 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ dom 𝑅1)
41 r1ord3g 8525 . . . . . . . . . . . . . . . . . 18 ((suc 𝑦 ∈ dom 𝑅1𝑥 ∈ dom 𝑅1) → (suc 𝑦𝑥 → (𝑅1‘suc 𝑦) ⊆ (𝑅1𝑥)))
4239, 40, 41syl2anc 691 . . . . . . . . . . . . . . . . 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 8546 . . . . . . . . . . . . . . . . . 18 (𝑦 (𝑅1 “ On) → 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)))
4544ad2antrl 760 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)))
46 suceq 5707 . . . . . . . . . . . . . . . . . . 19 ((rank‘𝑦) = 𝑦 → suc (rank‘𝑦) = suc 𝑦)
4746ad2antll 761 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc (rank‘𝑦) = suc 𝑦)
4847fveq2d 6107 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅1‘suc (rank‘𝑦)) = (𝑅1‘suc 𝑦))
4945, 48eleqtrd 2690 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘suc 𝑦))
5043, 49sseldd 3569 . . . . . . . . . . . . . . 15 (((𝑥 ∈ dom 𝑅1𝑦𝑥) ∧ (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1𝑥))
5150ex 449 . . . . . . . . . . . . . 14 ((𝑥 ∈ dom 𝑅1𝑦𝑥) → ((𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → 𝑦 ∈ (𝑅1𝑥)))
5251ralimdva 2945 . . . . . . . . . . . . 13 (𝑥 ∈ dom 𝑅1 → (∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ∀𝑦𝑥 𝑦 ∈ (𝑅1𝑥)))
5352imp 444 . . . . . . . . . . . 12 ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → ∀𝑦𝑥 𝑦 ∈ (𝑅1𝑥))
54 dfss3 3558 . . . . . . . . . . . 12 (𝑥 ⊆ (𝑅1𝑥) ↔ ∀𝑦𝑥 𝑦 ∈ (𝑅1𝑥))
5553, 54sylibr 223 . . . . . . . . . . 11 ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ⊆ (𝑅1𝑥))
56 vex 3176 . . . . . . . . . . . 12 𝑥 ∈ V
5756elpw 4114 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 (𝑅1𝑥) ↔ 𝑥 ⊆ (𝑅1𝑥))
5855, 57sylibr 223 . . . . . . . . . 10 ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ 𝒫 (𝑅1𝑥))
59 r1sucg 8515 . . . . . . . . . . 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 2691 . . . . . . . . 9 ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ (𝑅1‘suc 𝑥))
62 r1elwf 8542 . . . . . . . . 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 8572 . . . . . . . . . 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 2676 . . . . . . . . . . . . . . . 16 ((rank‘𝑦) = 𝑦 → ((rank‘𝑦) ∈ 𝑧𝑦𝑧))
6766adantl 481 . . . . . . . . . . . . . . 15 ((𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ((rank‘𝑦) ∈ 𝑧𝑦𝑧))
6867ralimi 2936 . . . . . . . . . . . . . 14 (∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ∀𝑦𝑥 ((rank‘𝑦) ∈ 𝑧𝑦𝑧))
69 ralbi 3050 . . . . . . . . . . . . . 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 3558 . . . . . . . . . . . . 13 (𝑥𝑧 ↔ ∀𝑦𝑥 𝑦𝑧)
7270, 71syl6bbr 277 . . . . . . . . . . . 12 (∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → (∀𝑦𝑥 (rank‘𝑦) ∈ 𝑧𝑥𝑧))
7372rabbidv 3164 . . . . . . . . . . 11 (∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → {𝑧 ∈ On ∣ ∀𝑦𝑥 (rank‘𝑦) ∈ 𝑧} = {𝑧 ∈ On ∣ 𝑥𝑧})
7473inteqd 4415 . . . . . . . . . 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 4432 . . . . . . . . . 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 2648 . . . . . . . 8 ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (rank‘𝑥) = 𝑥)
8063, 79jca 553 . . . . . . 7 ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑥 (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥))
8180ex 449 . . . . . 6 (𝑥 ∈ dom 𝑅1 → (∀𝑦𝑥 (𝑦 (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → (𝑥 (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)))
8226, 81sylbid 229 . . . . 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 6949 . 2 (𝐴 ∈ On → (𝐴 ∈ dom 𝑅1 → (𝐴 (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴)))
866, 85mpcom 37 1 (𝐴 ∈ dom 𝑅1 → (𝐴 (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ∩ cint 4410  dom cdm 5038   “ cima 5041  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642  Fun wfun 5798  ‘cfv 5804  𝑅1cr1 8508  rankcrnk 8509 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-r1 8510  df-rank 8511 This theorem is referenced by:  rankonid  8575  onwf  8576  onssr1  8577
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