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Theorem rankonidlem 9771
Description: Lemma for rankonid 9772. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.)
Assertion
Ref Expression
rankonidlem (𝐴 ∈ dom 𝑅1 β†’ (𝐴 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π΄) = 𝐴))

Proof of Theorem rankonidlem
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r1funlim 9709 . . . . 5 (Fun 𝑅1 ∧ Lim dom 𝑅1)
21simpri 487 . . . 4 Lim dom 𝑅1
3 limord 6382 . . . 4 (Lim dom 𝑅1 β†’ Ord dom 𝑅1)
42, 3ax-mp 5 . . 3 Ord dom 𝑅1
5 ordelon 6346 . . 3 ((Ord dom 𝑅1 ∧ 𝐴 ∈ dom 𝑅1) β†’ 𝐴 ∈ On)
64, 5mpan 689 . 2 (𝐴 ∈ dom 𝑅1 β†’ 𝐴 ∈ On)
7 eleq1 2826 . . . 4 (π‘₯ = 𝑦 β†’ (π‘₯ ∈ dom 𝑅1 ↔ 𝑦 ∈ dom 𝑅1))
8 eleq1 2826 . . . . 5 (π‘₯ = 𝑦 β†’ (π‘₯ ∈ βˆͺ (𝑅1 β€œ On) ↔ 𝑦 ∈ βˆͺ (𝑅1 β€œ On)))
9 fveq2 6847 . . . . . 6 (π‘₯ = 𝑦 β†’ (rankβ€˜π‘₯) = (rankβ€˜π‘¦))
10 id 22 . . . . . 6 (π‘₯ = 𝑦 β†’ π‘₯ = 𝑦)
119, 10eqeq12d 2753 . . . . 5 (π‘₯ = 𝑦 β†’ ((rankβ€˜π‘₯) = π‘₯ ↔ (rankβ€˜π‘¦) = 𝑦))
128, 11anbi12d 632 . . . 4 (π‘₯ = 𝑦 β†’ ((π‘₯ ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘₯) = π‘₯) ↔ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)))
137, 12imbi12d 345 . . 3 (π‘₯ = 𝑦 β†’ ((π‘₯ ∈ dom 𝑅1 β†’ (π‘₯ ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘₯) = π‘₯)) ↔ (𝑦 ∈ dom 𝑅1 β†’ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦))))
14 eleq1 2826 . . . 4 (π‘₯ = 𝐴 β†’ (π‘₯ ∈ dom 𝑅1 ↔ 𝐴 ∈ dom 𝑅1))
15 eleq1 2826 . . . . 5 (π‘₯ = 𝐴 β†’ (π‘₯ ∈ βˆͺ (𝑅1 β€œ On) ↔ 𝐴 ∈ βˆͺ (𝑅1 β€œ On)))
16 fveq2 6847 . . . . . 6 (π‘₯ = 𝐴 β†’ (rankβ€˜π‘₯) = (rankβ€˜π΄))
17 id 22 . . . . . 6 (π‘₯ = 𝐴 β†’ π‘₯ = 𝐴)
1816, 17eqeq12d 2753 . . . . 5 (π‘₯ = 𝐴 β†’ ((rankβ€˜π‘₯) = π‘₯ ↔ (rankβ€˜π΄) = 𝐴))
1915, 18anbi12d 632 . . . 4 (π‘₯ = 𝐴 β†’ ((π‘₯ ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘₯) = π‘₯) ↔ (𝐴 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π΄) = 𝐴)))
2014, 19imbi12d 345 . . 3 (π‘₯ = 𝐴 β†’ ((π‘₯ ∈ dom 𝑅1 β†’ (π‘₯ ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘₯) = π‘₯)) ↔ (𝐴 ∈ dom 𝑅1 β†’ (𝐴 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π΄) = 𝐴))))
21 ordtr1 6365 . . . . . . . . . 10 (Ord dom 𝑅1 β†’ ((𝑦 ∈ π‘₯ ∧ π‘₯ ∈ dom 𝑅1) β†’ 𝑦 ∈ dom 𝑅1))
224, 21ax-mp 5 . . . . . . . . 9 ((𝑦 ∈ π‘₯ ∧ π‘₯ ∈ dom 𝑅1) β†’ 𝑦 ∈ dom 𝑅1)
2322ancoms 460 . . . . . . . 8 ((π‘₯ ∈ dom 𝑅1 ∧ 𝑦 ∈ π‘₯) β†’ 𝑦 ∈ dom 𝑅1)
24 pm5.5 362 . . . . . . . 8 (𝑦 ∈ dom 𝑅1 β†’ ((𝑦 ∈ dom 𝑅1 β†’ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) ↔ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)))
2523, 24syl 17 . . . . . . 7 ((π‘₯ ∈ dom 𝑅1 ∧ 𝑦 ∈ π‘₯) β†’ ((𝑦 ∈ dom 𝑅1 β†’ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) ↔ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)))
2625ralbidva 3173 . . . . . 6 (π‘₯ ∈ dom 𝑅1 β†’ (βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ dom 𝑅1 β†’ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) ↔ βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)))
27 simplr 768 . . . . . . . . . . . . . . . . . 18 (((π‘₯ ∈ dom 𝑅1 ∧ 𝑦 ∈ π‘₯) ∧ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ 𝑦 ∈ π‘₯)
28 ordelon 6346 . . . . . . . . . . . . . . . . . . . . . 22 ((Ord dom 𝑅1 ∧ π‘₯ ∈ dom 𝑅1) β†’ π‘₯ ∈ On)
294, 28mpan 689 . . . . . . . . . . . . . . . . . . . . 21 (π‘₯ ∈ dom 𝑅1 β†’ π‘₯ ∈ On)
3029ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20 (((π‘₯ ∈ dom 𝑅1 ∧ 𝑦 ∈ π‘₯) ∧ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ π‘₯ ∈ On)
31 eloni 6332 . . . . . . . . . . . . . . . . . . . 20 (π‘₯ ∈ On β†’ Ord π‘₯)
3230, 31syl 17 . . . . . . . . . . . . . . . . . . 19 (((π‘₯ ∈ dom 𝑅1 ∧ 𝑦 ∈ π‘₯) ∧ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ Ord π‘₯)
33 ordelsuc 7760 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ π‘₯ ∧ Ord π‘₯) β†’ (𝑦 ∈ π‘₯ ↔ suc 𝑦 βŠ† π‘₯))
3427, 32, 33syl2anc 585 . . . . . . . . . . . . . . . . . 18 (((π‘₯ ∈ dom 𝑅1 ∧ 𝑦 ∈ π‘₯) ∧ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ (𝑦 ∈ π‘₯ ↔ suc 𝑦 βŠ† π‘₯))
3527, 34mpbid 231 . . . . . . . . . . . . . . . . 17 (((π‘₯ ∈ dom 𝑅1 ∧ 𝑦 ∈ π‘₯) ∧ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ suc 𝑦 βŠ† π‘₯)
3623adantr 482 . . . . . . . . . . . . . . . . . . 19 (((π‘₯ ∈ dom 𝑅1 ∧ 𝑦 ∈ π‘₯) ∧ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ 𝑦 ∈ dom 𝑅1)
37 limsuc 7790 . . . . . . . . . . . . . . . . . . . 20 (Lim dom 𝑅1 β†’ (𝑦 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1))
382, 37ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1)
3936, 38sylib 217 . . . . . . . . . . . . . . . . . 18 (((π‘₯ ∈ dom 𝑅1 ∧ 𝑦 ∈ π‘₯) ∧ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ suc 𝑦 ∈ dom 𝑅1)
40 simpll 766 . . . . . . . . . . . . . . . . . 18 (((π‘₯ ∈ dom 𝑅1 ∧ 𝑦 ∈ π‘₯) ∧ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ π‘₯ ∈ dom 𝑅1)
41 r1ord3g 9722 . . . . . . . . . . . . . . . . . 18 ((suc 𝑦 ∈ dom 𝑅1 ∧ π‘₯ ∈ dom 𝑅1) β†’ (suc 𝑦 βŠ† π‘₯ β†’ (𝑅1β€˜suc 𝑦) βŠ† (𝑅1β€˜π‘₯)))
4239, 40, 41syl2anc 585 . . . . . . . . . . . . . . . . 17 (((π‘₯ ∈ dom 𝑅1 ∧ 𝑦 ∈ π‘₯) ∧ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ (suc 𝑦 βŠ† π‘₯ β†’ (𝑅1β€˜suc 𝑦) βŠ† (𝑅1β€˜π‘₯)))
4335, 42mpd 15 . . . . . . . . . . . . . . . 16 (((π‘₯ ∈ dom 𝑅1 ∧ 𝑦 ∈ π‘₯) ∧ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ (𝑅1β€˜suc 𝑦) βŠ† (𝑅1β€˜π‘₯))
44 rankidb 9743 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ βˆͺ (𝑅1 β€œ On) β†’ 𝑦 ∈ (𝑅1β€˜suc (rankβ€˜π‘¦)))
4544ad2antrl 727 . . . . . . . . . . . . . . . . 17 (((π‘₯ ∈ dom 𝑅1 ∧ 𝑦 ∈ π‘₯) ∧ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ 𝑦 ∈ (𝑅1β€˜suc (rankβ€˜π‘¦)))
46 suceq 6388 . . . . . . . . . . . . . . . . . . 19 ((rankβ€˜π‘¦) = 𝑦 β†’ suc (rankβ€˜π‘¦) = suc 𝑦)
4746ad2antll 728 . . . . . . . . . . . . . . . . . 18 (((π‘₯ ∈ dom 𝑅1 ∧ 𝑦 ∈ π‘₯) ∧ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ suc (rankβ€˜π‘¦) = suc 𝑦)
4847fveq2d 6851 . . . . . . . . . . . . . . . . 17 (((π‘₯ ∈ dom 𝑅1 ∧ 𝑦 ∈ π‘₯) ∧ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ (𝑅1β€˜suc (rankβ€˜π‘¦)) = (𝑅1β€˜suc 𝑦))
4945, 48eleqtrd 2840 . . . . . . . . . . . . . . . 16 (((π‘₯ ∈ dom 𝑅1 ∧ 𝑦 ∈ π‘₯) ∧ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ 𝑦 ∈ (𝑅1β€˜suc 𝑦))
5043, 49sseldd 3950 . . . . . . . . . . . . . . 15 (((π‘₯ ∈ dom 𝑅1 ∧ 𝑦 ∈ π‘₯) ∧ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ 𝑦 ∈ (𝑅1β€˜π‘₯))
5150ex 414 . . . . . . . . . . . . . 14 ((π‘₯ ∈ dom 𝑅1 ∧ 𝑦 ∈ π‘₯) β†’ ((𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦) β†’ 𝑦 ∈ (𝑅1β€˜π‘₯)))
5251ralimdva 3165 . . . . . . . . . . . . 13 (π‘₯ ∈ dom 𝑅1 β†’ (βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦) β†’ βˆ€π‘¦ ∈ π‘₯ 𝑦 ∈ (𝑅1β€˜π‘₯)))
5352imp 408 . . . . . . . . . . . 12 ((π‘₯ ∈ dom 𝑅1 ∧ βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ βˆ€π‘¦ ∈ π‘₯ 𝑦 ∈ (𝑅1β€˜π‘₯))
54 dfss3 3937 . . . . . . . . . . . 12 (π‘₯ βŠ† (𝑅1β€˜π‘₯) ↔ βˆ€π‘¦ ∈ π‘₯ 𝑦 ∈ (𝑅1β€˜π‘₯))
5553, 54sylibr 233 . . . . . . . . . . 11 ((π‘₯ ∈ dom 𝑅1 ∧ βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ π‘₯ βŠ† (𝑅1β€˜π‘₯))
56 vex 3452 . . . . . . . . . . . 12 π‘₯ ∈ V
5756elpw 4569 . . . . . . . . . . 11 (π‘₯ ∈ 𝒫 (𝑅1β€˜π‘₯) ↔ π‘₯ βŠ† (𝑅1β€˜π‘₯))
5855, 57sylibr 233 . . . . . . . . . 10 ((π‘₯ ∈ dom 𝑅1 ∧ βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ π‘₯ ∈ 𝒫 (𝑅1β€˜π‘₯))
59 r1sucg 9712 . . . . . . . . . . 11 (π‘₯ ∈ dom 𝑅1 β†’ (𝑅1β€˜suc π‘₯) = 𝒫 (𝑅1β€˜π‘₯))
6059adantr 482 . . . . . . . . . 10 ((π‘₯ ∈ dom 𝑅1 ∧ βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ (𝑅1β€˜suc π‘₯) = 𝒫 (𝑅1β€˜π‘₯))
6158, 60eleqtrrd 2841 . . . . . . . . 9 ((π‘₯ ∈ dom 𝑅1 ∧ βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ π‘₯ ∈ (𝑅1β€˜suc π‘₯))
62 r1elwf 9739 . . . . . . . . 9 (π‘₯ ∈ (𝑅1β€˜suc π‘₯) β†’ π‘₯ ∈ βˆͺ (𝑅1 β€œ On))
6361, 62syl 17 . . . . . . . 8 ((π‘₯ ∈ dom 𝑅1 ∧ βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ π‘₯ ∈ βˆͺ (𝑅1 β€œ On))
64 rankval3b 9769 . . . . . . . . . 10 (π‘₯ ∈ βˆͺ (𝑅1 β€œ On) β†’ (rankβ€˜π‘₯) = ∩ {𝑧 ∈ On ∣ βˆ€π‘¦ ∈ π‘₯ (rankβ€˜π‘¦) ∈ 𝑧})
6563, 64syl 17 . . . . . . . . 9 ((π‘₯ ∈ dom 𝑅1 ∧ βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ (rankβ€˜π‘₯) = ∩ {𝑧 ∈ On ∣ βˆ€π‘¦ ∈ π‘₯ (rankβ€˜π‘¦) ∈ 𝑧})
66 eleq1 2826 . . . . . . . . . . . . . . . 16 ((rankβ€˜π‘¦) = 𝑦 β†’ ((rankβ€˜π‘¦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
6766adantl 483 . . . . . . . . . . . . . . 15 ((𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦) β†’ ((rankβ€˜π‘¦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
6867ralimi 3087 . . . . . . . . . . . . . 14 (βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦) β†’ βˆ€π‘¦ ∈ π‘₯ ((rankβ€˜π‘¦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
69 ralbi 3107 . . . . . . . . . . . . . 14 (βˆ€π‘¦ ∈ π‘₯ ((rankβ€˜π‘¦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧) β†’ (βˆ€π‘¦ ∈ π‘₯ (rankβ€˜π‘¦) ∈ 𝑧 ↔ βˆ€π‘¦ ∈ π‘₯ 𝑦 ∈ 𝑧))
7068, 69syl 17 . . . . . . . . . . . . 13 (βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦) β†’ (βˆ€π‘¦ ∈ π‘₯ (rankβ€˜π‘¦) ∈ 𝑧 ↔ βˆ€π‘¦ ∈ π‘₯ 𝑦 ∈ 𝑧))
71 dfss3 3937 . . . . . . . . . . . . 13 (π‘₯ βŠ† 𝑧 ↔ βˆ€π‘¦ ∈ π‘₯ 𝑦 ∈ 𝑧)
7270, 71bitr4di 289 . . . . . . . . . . . 12 (βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦) β†’ (βˆ€π‘¦ ∈ π‘₯ (rankβ€˜π‘¦) ∈ 𝑧 ↔ π‘₯ βŠ† 𝑧))
7372rabbidv 3418 . . . . . . . . . . 11 (βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦) β†’ {𝑧 ∈ On ∣ βˆ€π‘¦ ∈ π‘₯ (rankβ€˜π‘¦) ∈ 𝑧} = {𝑧 ∈ On ∣ π‘₯ βŠ† 𝑧})
7473inteqd 4917 . . . . . . . . . 10 (βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦) β†’ ∩ {𝑧 ∈ On ∣ βˆ€π‘¦ ∈ π‘₯ (rankβ€˜π‘¦) ∈ 𝑧} = ∩ {𝑧 ∈ On ∣ π‘₯ βŠ† 𝑧})
7574adantl 483 . . . . . . . . 9 ((π‘₯ ∈ dom 𝑅1 ∧ βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ ∩ {𝑧 ∈ On ∣ βˆ€π‘¦ ∈ π‘₯ (rankβ€˜π‘¦) ∈ 𝑧} = ∩ {𝑧 ∈ On ∣ π‘₯ βŠ† 𝑧})
7629adantr 482 . . . . . . . . . 10 ((π‘₯ ∈ dom 𝑅1 ∧ βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ π‘₯ ∈ On)
77 intmin 4934 . . . . . . . . . 10 (π‘₯ ∈ On β†’ ∩ {𝑧 ∈ On ∣ π‘₯ βŠ† 𝑧} = π‘₯)
7876, 77syl 17 . . . . . . . . 9 ((π‘₯ ∈ dom 𝑅1 ∧ βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ ∩ {𝑧 ∈ On ∣ π‘₯ βŠ† 𝑧} = π‘₯)
7965, 75, 783eqtrd 2781 . . . . . . . 8 ((π‘₯ ∈ dom 𝑅1 ∧ βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ (rankβ€˜π‘₯) = π‘₯)
8063, 79jca 513 . . . . . . 7 ((π‘₯ ∈ dom 𝑅1 ∧ βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ (π‘₯ ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘₯) = π‘₯))
8180ex 414 . . . . . 6 (π‘₯ ∈ dom 𝑅1 β†’ (βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦) β†’ (π‘₯ ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘₯) = π‘₯)))
8226, 81sylbid 239 . . . . 5 (π‘₯ ∈ dom 𝑅1 β†’ (βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ dom 𝑅1 β†’ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ (π‘₯ ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘₯) = π‘₯)))
8382com12 32 . . . 4 (βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ dom 𝑅1 β†’ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ (π‘₯ ∈ dom 𝑅1 β†’ (π‘₯ ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘₯) = π‘₯)))
8483a1i 11 . . 3 (π‘₯ ∈ On β†’ (βˆ€π‘¦ ∈ π‘₯ (𝑦 ∈ dom 𝑅1 β†’ (𝑦 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘¦) = 𝑦)) β†’ (π‘₯ ∈ dom 𝑅1 β†’ (π‘₯ ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π‘₯) = π‘₯))))
8513, 20, 84tfis3 7799 . 2 (𝐴 ∈ On β†’ (𝐴 ∈ dom 𝑅1 β†’ (𝐴 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π΄) = 𝐴)))
866, 85mpcom 38 1 (𝐴 ∈ dom 𝑅1 β†’ (𝐴 ∈ βˆͺ (𝑅1 β€œ On) ∧ (rankβ€˜π΄) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3410   βŠ† wss 3915  π’« cpw 4565  βˆͺ cuni 4870  βˆ© cint 4912  dom cdm 5638   β€œ cima 5641  Ord word 6321  Oncon0 6322  Lim wlim 6323  suc csuc 6324  Fun wfun 6495  β€˜cfv 6501  π‘…1cr1 9705  rankcrnk 9706
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 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-r1 9707  df-rank 9708
This theorem is referenced by:  rankonid  9772  onwf  9773  onssr1  9774
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