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Theorem ackbij2lem3 10238
Description: Lemma for ackbij2 10240. (Contributed by Stefan O'Rear, 18-Nov-2014.)
Hypotheses
Ref Expression
ackbij.f 𝐹 = (π‘₯ ∈ (𝒫 Ο‰ ∩ Fin) ↦ (cardβ€˜βˆͺ 𝑦 ∈ π‘₯ ({𝑦} Γ— 𝒫 𝑦)))
ackbij.g 𝐺 = (π‘₯ ∈ V ↦ (𝑦 ∈ 𝒫 dom π‘₯ ↦ (πΉβ€˜(π‘₯ β€œ 𝑦))))
Assertion
Ref Expression
ackbij2lem3 (𝐴 ∈ Ο‰ β†’ (rec(𝐺, βˆ…)β€˜π΄) βŠ† (rec(𝐺, βˆ…)β€˜suc 𝐴))
Distinct variable groups:   π‘₯,𝐹,𝑦   π‘₯,𝐺,𝑦   π‘₯,𝐴,𝑦
Proof of Theorem ackbij2lem3
Dummy variables π‘Ž 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6891 . . . 4 (π‘Ž = βˆ… β†’ (rec(𝐺, βˆ…)β€˜π‘Ž) = (rec(𝐺, βˆ…)β€˜βˆ…))
2 suceq 6430 . . . . . 6 (π‘Ž = βˆ… β†’ suc π‘Ž = suc βˆ…)
32fveq2d 6895 . . . . 5 (π‘Ž = βˆ… β†’ (rec(𝐺, βˆ…)β€˜suc π‘Ž) = (rec(𝐺, βˆ…)β€˜suc βˆ…))
4 fveq2 6891 . . . . 5 (π‘Ž = βˆ… β†’ (𝑅1β€˜π‘Ž) = (𝑅1β€˜βˆ…))
53, 4reseq12d 5982 . . . 4 (π‘Ž = βˆ… β†’ ((rec(𝐺, βˆ…)β€˜suc π‘Ž) β†Ύ (𝑅1β€˜π‘Ž)) = ((rec(𝐺, βˆ…)β€˜suc βˆ…) β†Ύ (𝑅1β€˜βˆ…)))
61, 5eqeq12d 2748 . . 3 (π‘Ž = βˆ… β†’ ((rec(𝐺, βˆ…)β€˜π‘Ž) = ((rec(𝐺, βˆ…)β€˜suc π‘Ž) β†Ύ (𝑅1β€˜π‘Ž)) ↔ (rec(𝐺, βˆ…)β€˜βˆ…) = ((rec(𝐺, βˆ…)β€˜suc βˆ…) β†Ύ (𝑅1β€˜βˆ…))))
7 fveq2 6891 . . . 4 (π‘Ž = 𝑏 β†’ (rec(𝐺, βˆ…)β€˜π‘Ž) = (rec(𝐺, βˆ…)β€˜π‘))
8 suceq 6430 . . . . . 6 (π‘Ž = 𝑏 β†’ suc π‘Ž = suc 𝑏)
98fveq2d 6895 . . . . 5 (π‘Ž = 𝑏 β†’ (rec(𝐺, βˆ…)β€˜suc π‘Ž) = (rec(𝐺, βˆ…)β€˜suc 𝑏))
10 fveq2 6891 . . . . 5 (π‘Ž = 𝑏 β†’ (𝑅1β€˜π‘Ž) = (𝑅1β€˜π‘))
119, 10reseq12d 5982 . . . 4 (π‘Ž = 𝑏 β†’ ((rec(𝐺, βˆ…)β€˜suc π‘Ž) β†Ύ (𝑅1β€˜π‘Ž)) = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)))
127, 11eqeq12d 2748 . . 3 (π‘Ž = 𝑏 β†’ ((rec(𝐺, βˆ…)β€˜π‘Ž) = ((rec(𝐺, βˆ…)β€˜suc π‘Ž) β†Ύ (𝑅1β€˜π‘Ž)) ↔ (rec(𝐺, βˆ…)β€˜π‘) = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘))))
13 fveq2 6891 . . . 4 (π‘Ž = suc 𝑏 β†’ (rec(𝐺, βˆ…)β€˜π‘Ž) = (rec(𝐺, βˆ…)β€˜suc 𝑏))
14 suceq 6430 . . . . . 6 (π‘Ž = suc 𝑏 β†’ suc π‘Ž = suc suc 𝑏)
1514fveq2d 6895 . . . . 5 (π‘Ž = suc 𝑏 β†’ (rec(𝐺, βˆ…)β€˜suc π‘Ž) = (rec(𝐺, βˆ…)β€˜suc suc 𝑏))
16 fveq2 6891 . . . . 5 (π‘Ž = suc 𝑏 β†’ (𝑅1β€˜π‘Ž) = (𝑅1β€˜suc 𝑏))
1715, 16reseq12d 5982 . . . 4 (π‘Ž = suc 𝑏 β†’ ((rec(𝐺, βˆ…)β€˜suc π‘Ž) β†Ύ (𝑅1β€˜π‘Ž)) = ((rec(𝐺, βˆ…)β€˜suc suc 𝑏) β†Ύ (𝑅1β€˜suc 𝑏)))
1813, 17eqeq12d 2748 . . 3 (π‘Ž = suc 𝑏 β†’ ((rec(𝐺, βˆ…)β€˜π‘Ž) = ((rec(𝐺, βˆ…)β€˜suc π‘Ž) β†Ύ (𝑅1β€˜π‘Ž)) ↔ (rec(𝐺, βˆ…)β€˜suc 𝑏) = ((rec(𝐺, βˆ…)β€˜suc suc 𝑏) β†Ύ (𝑅1β€˜suc 𝑏))))
19 fveq2 6891 . . . 4 (π‘Ž = 𝐴 β†’ (rec(𝐺, βˆ…)β€˜π‘Ž) = (rec(𝐺, βˆ…)β€˜π΄))
20 suceq 6430 . . . . . 6 (π‘Ž = 𝐴 β†’ suc π‘Ž = suc 𝐴)
2120fveq2d 6895 . . . . 5 (π‘Ž = 𝐴 β†’ (rec(𝐺, βˆ…)β€˜suc π‘Ž) = (rec(𝐺, βˆ…)β€˜suc 𝐴))
22 fveq2 6891 . . . . 5 (π‘Ž = 𝐴 β†’ (𝑅1β€˜π‘Ž) = (𝑅1β€˜π΄))
2321, 22reseq12d 5982 . . . 4 (π‘Ž = 𝐴 β†’ ((rec(𝐺, βˆ…)β€˜suc π‘Ž) β†Ύ (𝑅1β€˜π‘Ž)) = ((rec(𝐺, βˆ…)β€˜suc 𝐴) β†Ύ (𝑅1β€˜π΄)))
2419, 23eqeq12d 2748 . . 3 (π‘Ž = 𝐴 β†’ ((rec(𝐺, βˆ…)β€˜π‘Ž) = ((rec(𝐺, βˆ…)β€˜suc π‘Ž) β†Ύ (𝑅1β€˜π‘Ž)) ↔ (rec(𝐺, βˆ…)β€˜π΄) = ((rec(𝐺, βˆ…)β€˜suc 𝐴) β†Ύ (𝑅1β€˜π΄))))
25 res0 5985 . . . 4 ((rec(𝐺, βˆ…)β€˜suc βˆ…) β†Ύ βˆ…) = βˆ…
26 r10 9765 . . . . 5 (𝑅1β€˜βˆ…) = βˆ…
2726reseq2i 5978 . . . 4 ((rec(𝐺, βˆ…)β€˜suc βˆ…) β†Ύ (𝑅1β€˜βˆ…)) = ((rec(𝐺, βˆ…)β€˜suc βˆ…) β†Ύ βˆ…)
28 0ex 5307 . . . . 5 βˆ… ∈ V
2928rdg0 8423 . . . 4 (rec(𝐺, βˆ…)β€˜βˆ…) = βˆ…
3025, 27, 293eqtr4ri 2771 . . 3 (rec(𝐺, βˆ…)β€˜βˆ…) = ((rec(𝐺, βˆ…)β€˜suc βˆ…) β†Ύ (𝑅1β€˜βˆ…))
31 peano2 7883 . . . . . . . 8 (𝑏 ∈ Ο‰ β†’ suc 𝑏 ∈ Ο‰)
32 ackbij.f . . . . . . . . 9 𝐹 = (π‘₯ ∈ (𝒫 Ο‰ ∩ Fin) ↦ (cardβ€˜βˆͺ 𝑦 ∈ π‘₯ ({𝑦} Γ— 𝒫 𝑦)))
33 ackbij.g . . . . . . . . 9 𝐺 = (π‘₯ ∈ V ↦ (𝑦 ∈ 𝒫 dom π‘₯ ↦ (πΉβ€˜(π‘₯ β€œ 𝑦))))
3432, 33ackbij2lem2 10237 . . . . . . . 8 (suc 𝑏 ∈ Ο‰ β†’ (rec(𝐺, βˆ…)β€˜suc 𝑏):(𝑅1β€˜suc 𝑏)–1-1-ontoβ†’(cardβ€˜(𝑅1β€˜suc 𝑏)))
3531, 34syl 17 . . . . . . 7 (𝑏 ∈ Ο‰ β†’ (rec(𝐺, βˆ…)β€˜suc 𝑏):(𝑅1β€˜suc 𝑏)–1-1-ontoβ†’(cardβ€˜(𝑅1β€˜suc 𝑏)))
36 f1ofn 6834 . . . . . . 7 ((rec(𝐺, βˆ…)β€˜suc 𝑏):(𝑅1β€˜suc 𝑏)–1-1-ontoβ†’(cardβ€˜(𝑅1β€˜suc 𝑏)) β†’ (rec(𝐺, βˆ…)β€˜suc 𝑏) Fn (𝑅1β€˜suc 𝑏))
3735, 36syl 17 . . . . . 6 (𝑏 ∈ Ο‰ β†’ (rec(𝐺, βˆ…)β€˜suc 𝑏) Fn (𝑅1β€˜suc 𝑏))
3837adantr 481 . . . . 5 ((𝑏 ∈ Ο‰ ∧ (rec(𝐺, βˆ…)β€˜π‘) = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘))) β†’ (rec(𝐺, βˆ…)β€˜suc 𝑏) Fn (𝑅1β€˜suc 𝑏))
39 peano2 7883 . . . . . . . 8 (suc 𝑏 ∈ Ο‰ β†’ suc suc 𝑏 ∈ Ο‰)
4032, 33ackbij2lem2 10237 . . . . . . . 8 (suc suc 𝑏 ∈ Ο‰ β†’ (rec(𝐺, βˆ…)β€˜suc suc 𝑏):(𝑅1β€˜suc suc 𝑏)–1-1-ontoβ†’(cardβ€˜(𝑅1β€˜suc suc 𝑏)))
41 f1ofn 6834 . . . . . . . 8 ((rec(𝐺, βˆ…)β€˜suc suc 𝑏):(𝑅1β€˜suc suc 𝑏)–1-1-ontoβ†’(cardβ€˜(𝑅1β€˜suc suc 𝑏)) β†’ (rec(𝐺, βˆ…)β€˜suc suc 𝑏) Fn (𝑅1β€˜suc suc 𝑏))
4231, 39, 40, 414syl 19 . . . . . . 7 (𝑏 ∈ Ο‰ β†’ (rec(𝐺, βˆ…)β€˜suc suc 𝑏) Fn (𝑅1β€˜suc suc 𝑏))
43 nnon 7863 . . . . . . . . 9 (suc 𝑏 ∈ Ο‰ β†’ suc 𝑏 ∈ On)
4431, 43syl 17 . . . . . . . 8 (𝑏 ∈ Ο‰ β†’ suc 𝑏 ∈ On)
45 r1sssuc 9780 . . . . . . . 8 (suc 𝑏 ∈ On β†’ (𝑅1β€˜suc 𝑏) βŠ† (𝑅1β€˜suc suc 𝑏))
4644, 45syl 17 . . . . . . 7 (𝑏 ∈ Ο‰ β†’ (𝑅1β€˜suc 𝑏) βŠ† (𝑅1β€˜suc suc 𝑏))
47 fnssres 6673 . . . . . . 7 (((rec(𝐺, βˆ…)β€˜suc suc 𝑏) Fn (𝑅1β€˜suc suc 𝑏) ∧ (𝑅1β€˜suc 𝑏) βŠ† (𝑅1β€˜suc suc 𝑏)) β†’ ((rec(𝐺, βˆ…)β€˜suc suc 𝑏) β†Ύ (𝑅1β€˜suc 𝑏)) Fn (𝑅1β€˜suc 𝑏))
4842, 46, 47syl2anc 584 . . . . . 6 (𝑏 ∈ Ο‰ β†’ ((rec(𝐺, βˆ…)β€˜suc suc 𝑏) β†Ύ (𝑅1β€˜suc 𝑏)) Fn (𝑅1β€˜suc 𝑏))
4948adantr 481 . . . . 5 ((𝑏 ∈ Ο‰ ∧ (rec(𝐺, βˆ…)β€˜π‘) = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘))) β†’ ((rec(𝐺, βˆ…)β€˜suc suc 𝑏) β†Ύ (𝑅1β€˜suc 𝑏)) Fn (𝑅1β€˜suc 𝑏))
50 nnon 7863 . . . . . . . . . . . . . . 15 (𝑏 ∈ Ο‰ β†’ 𝑏 ∈ On)
51 r1suc 9767 . . . . . . . . . . . . . . 15 (𝑏 ∈ On β†’ (𝑅1β€˜suc 𝑏) = 𝒫 (𝑅1β€˜π‘))
5250, 51syl 17 . . . . . . . . . . . . . 14 (𝑏 ∈ Ο‰ β†’ (𝑅1β€˜suc 𝑏) = 𝒫 (𝑅1β€˜π‘))
5352eleq2d 2819 . . . . . . . . . . . . 13 (𝑏 ∈ Ο‰ β†’ (𝑐 ∈ (𝑅1β€˜suc 𝑏) ↔ 𝑐 ∈ 𝒫 (𝑅1β€˜π‘)))
5453biimpa 477 . . . . . . . . . . . 12 ((𝑏 ∈ Ο‰ ∧ 𝑐 ∈ (𝑅1β€˜suc 𝑏)) β†’ 𝑐 ∈ 𝒫 (𝑅1β€˜π‘))
5554elpwid 4611 . . . . . . . . . . 11 ((𝑏 ∈ Ο‰ ∧ 𝑐 ∈ (𝑅1β€˜suc 𝑏)) β†’ 𝑐 βŠ† (𝑅1β€˜π‘))
56 resima2 6016 . . . . . . . . . . 11 (𝑐 βŠ† (𝑅1β€˜π‘) β†’ (((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β€œ 𝑐) = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β€œ 𝑐))
5755, 56syl 17 . . . . . . . . . 10 ((𝑏 ∈ Ο‰ ∧ 𝑐 ∈ (𝑅1β€˜suc 𝑏)) β†’ (((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β€œ 𝑐) = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β€œ 𝑐))
5857fveq2d 6895 . . . . . . . . 9 ((𝑏 ∈ Ο‰ ∧ 𝑐 ∈ (𝑅1β€˜suc 𝑏)) β†’ (πΉβ€˜(((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β€œ 𝑐)) = (πΉβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β€œ 𝑐)))
59 fvex 6904 . . . . . . . . . . . . 13 (rec(𝐺, βˆ…)β€˜suc 𝑏) ∈ V
6059resex 6029 . . . . . . . . . . . 12 ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) ∈ V
61 dmeq 5903 . . . . . . . . . . . . . . 15 (π‘₯ = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β†’ dom π‘₯ = dom ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)))
6261pweqd 4619 . . . . . . . . . . . . . 14 (π‘₯ = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β†’ 𝒫 dom π‘₯ = 𝒫 dom ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)))
63 imaeq1 6054 . . . . . . . . . . . . . . 15 (π‘₯ = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β†’ (π‘₯ β€œ 𝑦) = (((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β€œ 𝑦))
6463fveq2d 6895 . . . . . . . . . . . . . 14 (π‘₯ = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β†’ (πΉβ€˜(π‘₯ β€œ 𝑦)) = (πΉβ€˜(((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β€œ 𝑦)))
6562, 64mpteq12dv 5239 . . . . . . . . . . . . 13 (π‘₯ = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β†’ (𝑦 ∈ 𝒫 dom π‘₯ ↦ (πΉβ€˜(π‘₯ β€œ 𝑦))) = (𝑦 ∈ 𝒫 dom ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) ↦ (πΉβ€˜(((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β€œ 𝑦))))
6660dmex 7904 . . . . . . . . . . . . . . 15 dom ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) ∈ V
6766pwex 5378 . . . . . . . . . . . . . 14 𝒫 dom ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) ∈ V
6867mptex 7227 . . . . . . . . . . . . 13 (𝑦 ∈ 𝒫 dom ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) ↦ (πΉβ€˜(((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β€œ 𝑦))) ∈ V
6965, 33, 68fvmpt 6998 . . . . . . . . . . . 12 (((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) ∈ V β†’ (πΊβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘))) = (𝑦 ∈ 𝒫 dom ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) ↦ (πΉβ€˜(((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β€œ 𝑦))))
7060, 69ax-mp 5 . . . . . . . . . . 11 (πΊβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘))) = (𝑦 ∈ 𝒫 dom ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) ↦ (πΉβ€˜(((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β€œ 𝑦)))
7170fveq1i 6892 . . . . . . . . . 10 ((πΊβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)))β€˜π‘) = ((𝑦 ∈ 𝒫 dom ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) ↦ (πΉβ€˜(((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β€œ 𝑦)))β€˜π‘)
72 r1sssuc 9780 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ On β†’ (𝑅1β€˜π‘) βŠ† (𝑅1β€˜suc 𝑏))
7350, 72syl 17 . . . . . . . . . . . . . . . 16 (𝑏 ∈ Ο‰ β†’ (𝑅1β€˜π‘) βŠ† (𝑅1β€˜suc 𝑏))
74 fnssres 6673 . . . . . . . . . . . . . . . 16 (((rec(𝐺, βˆ…)β€˜suc 𝑏) Fn (𝑅1β€˜suc 𝑏) ∧ (𝑅1β€˜π‘) βŠ† (𝑅1β€˜suc 𝑏)) β†’ ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) Fn (𝑅1β€˜π‘))
7537, 73, 74syl2anc 584 . . . . . . . . . . . . . . 15 (𝑏 ∈ Ο‰ β†’ ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) Fn (𝑅1β€˜π‘))
7675fndmd 6654 . . . . . . . . . . . . . 14 (𝑏 ∈ Ο‰ β†’ dom ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) = (𝑅1β€˜π‘))
7776pweqd 4619 . . . . . . . . . . . . 13 (𝑏 ∈ Ο‰ β†’ 𝒫 dom ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) = 𝒫 (𝑅1β€˜π‘))
7877adantr 481 . . . . . . . . . . . 12 ((𝑏 ∈ Ο‰ ∧ 𝑐 ∈ (𝑅1β€˜suc 𝑏)) β†’ 𝒫 dom ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) = 𝒫 (𝑅1β€˜π‘))
7954, 78eleqtrrd 2836 . . . . . . . . . . 11 ((𝑏 ∈ Ο‰ ∧ 𝑐 ∈ (𝑅1β€˜suc 𝑏)) β†’ 𝑐 ∈ 𝒫 dom ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)))
80 imaeq2 6055 . . . . . . . . . . . . 13 (𝑦 = 𝑐 β†’ (((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β€œ 𝑦) = (((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β€œ 𝑐))
8180fveq2d 6895 . . . . . . . . . . . 12 (𝑦 = 𝑐 β†’ (πΉβ€˜(((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β€œ 𝑦)) = (πΉβ€˜(((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β€œ 𝑐)))
82 eqid 2732 . . . . . . . . . . . 12 (𝑦 ∈ 𝒫 dom ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) ↦ (πΉβ€˜(((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β€œ 𝑦))) = (𝑦 ∈ 𝒫 dom ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) ↦ (πΉβ€˜(((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β€œ 𝑦)))
83 fvex 6904 . . . . . . . . . . . 12 (πΉβ€˜(((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β€œ 𝑐)) ∈ V
8481, 82, 83fvmpt 6998 . . . . . . . . . . 11 (𝑐 ∈ 𝒫 dom ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β†’ ((𝑦 ∈ 𝒫 dom ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) ↦ (πΉβ€˜(((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β€œ 𝑦)))β€˜π‘) = (πΉβ€˜(((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β€œ 𝑐)))
8579, 84syl 17 . . . . . . . . . 10 ((𝑏 ∈ Ο‰ ∧ 𝑐 ∈ (𝑅1β€˜suc 𝑏)) β†’ ((𝑦 ∈ 𝒫 dom ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) ↦ (πΉβ€˜(((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β€œ 𝑦)))β€˜π‘) = (πΉβ€˜(((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β€œ 𝑐)))
8671, 85eqtrid 2784 . . . . . . . . 9 ((𝑏 ∈ Ο‰ ∧ 𝑐 ∈ (𝑅1β€˜suc 𝑏)) β†’ ((πΊβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)))β€˜π‘) = (πΉβ€˜(((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β€œ 𝑐)))
87 dmeq 5903 . . . . . . . . . . . . . . 15 (π‘₯ = (rec(𝐺, βˆ…)β€˜suc 𝑏) β†’ dom π‘₯ = dom (rec(𝐺, βˆ…)β€˜suc 𝑏))
8887pweqd 4619 . . . . . . . . . . . . . 14 (π‘₯ = (rec(𝐺, βˆ…)β€˜suc 𝑏) β†’ 𝒫 dom π‘₯ = 𝒫 dom (rec(𝐺, βˆ…)β€˜suc 𝑏))
89 imaeq1 6054 . . . . . . . . . . . . . . 15 (π‘₯ = (rec(𝐺, βˆ…)β€˜suc 𝑏) β†’ (π‘₯ β€œ 𝑦) = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β€œ 𝑦))
9089fveq2d 6895 . . . . . . . . . . . . . 14 (π‘₯ = (rec(𝐺, βˆ…)β€˜suc 𝑏) β†’ (πΉβ€˜(π‘₯ β€œ 𝑦)) = (πΉβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β€œ 𝑦)))
9188, 90mpteq12dv 5239 . . . . . . . . . . . . 13 (π‘₯ = (rec(𝐺, βˆ…)β€˜suc 𝑏) β†’ (𝑦 ∈ 𝒫 dom π‘₯ ↦ (πΉβ€˜(π‘₯ β€œ 𝑦))) = (𝑦 ∈ 𝒫 dom (rec(𝐺, βˆ…)β€˜suc 𝑏) ↦ (πΉβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β€œ 𝑦))))
9259dmex 7904 . . . . . . . . . . . . . . 15 dom (rec(𝐺, βˆ…)β€˜suc 𝑏) ∈ V
9392pwex 5378 . . . . . . . . . . . . . 14 𝒫 dom (rec(𝐺, βˆ…)β€˜suc 𝑏) ∈ V
9493mptex 7227 . . . . . . . . . . . . 13 (𝑦 ∈ 𝒫 dom (rec(𝐺, βˆ…)β€˜suc 𝑏) ↦ (πΉβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β€œ 𝑦))) ∈ V
9591, 33, 94fvmpt 6998 . . . . . . . . . . . 12 ((rec(𝐺, βˆ…)β€˜suc 𝑏) ∈ V β†’ (πΊβ€˜(rec(𝐺, βˆ…)β€˜suc 𝑏)) = (𝑦 ∈ 𝒫 dom (rec(𝐺, βˆ…)β€˜suc 𝑏) ↦ (πΉβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β€œ 𝑦))))
9659, 95ax-mp 5 . . . . . . . . . . 11 (πΊβ€˜(rec(𝐺, βˆ…)β€˜suc 𝑏)) = (𝑦 ∈ 𝒫 dom (rec(𝐺, βˆ…)β€˜suc 𝑏) ↦ (πΉβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β€œ 𝑦)))
9796fveq1i 6892 . . . . . . . . . 10 ((πΊβ€˜(rec(𝐺, βˆ…)β€˜suc 𝑏))β€˜π‘) = ((𝑦 ∈ 𝒫 dom (rec(𝐺, βˆ…)β€˜suc 𝑏) ↦ (πΉβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β€œ 𝑦)))β€˜π‘)
98 r1tr 9773 . . . . . . . . . . . . . . 15 Tr (𝑅1β€˜suc 𝑏)
9998a1i 11 . . . . . . . . . . . . . 14 (𝑏 ∈ Ο‰ β†’ Tr (𝑅1β€˜suc 𝑏))
100 dftr4 5272 . . . . . . . . . . . . . 14 (Tr (𝑅1β€˜suc 𝑏) ↔ (𝑅1β€˜suc 𝑏) βŠ† 𝒫 (𝑅1β€˜suc 𝑏))
10199, 100sylib 217 . . . . . . . . . . . . 13 (𝑏 ∈ Ο‰ β†’ (𝑅1β€˜suc 𝑏) βŠ† 𝒫 (𝑅1β€˜suc 𝑏))
102101sselda 3982 . . . . . . . . . . . 12 ((𝑏 ∈ Ο‰ ∧ 𝑐 ∈ (𝑅1β€˜suc 𝑏)) β†’ 𝑐 ∈ 𝒫 (𝑅1β€˜suc 𝑏))
103 f1odm 6837 . . . . . . . . . . . . . . 15 ((rec(𝐺, βˆ…)β€˜suc 𝑏):(𝑅1β€˜suc 𝑏)–1-1-ontoβ†’(cardβ€˜(𝑅1β€˜suc 𝑏)) β†’ dom (rec(𝐺, βˆ…)β€˜suc 𝑏) = (𝑅1β€˜suc 𝑏))
10435, 103syl 17 . . . . . . . . . . . . . 14 (𝑏 ∈ Ο‰ β†’ dom (rec(𝐺, βˆ…)β€˜suc 𝑏) = (𝑅1β€˜suc 𝑏))
105104pweqd 4619 . . . . . . . . . . . . 13 (𝑏 ∈ Ο‰ β†’ 𝒫 dom (rec(𝐺, βˆ…)β€˜suc 𝑏) = 𝒫 (𝑅1β€˜suc 𝑏))
106105adantr 481 . . . . . . . . . . . 12 ((𝑏 ∈ Ο‰ ∧ 𝑐 ∈ (𝑅1β€˜suc 𝑏)) β†’ 𝒫 dom (rec(𝐺, βˆ…)β€˜suc 𝑏) = 𝒫 (𝑅1β€˜suc 𝑏))
107102, 106eleqtrrd 2836 . . . . . . . . . . 11 ((𝑏 ∈ Ο‰ ∧ 𝑐 ∈ (𝑅1β€˜suc 𝑏)) β†’ 𝑐 ∈ 𝒫 dom (rec(𝐺, βˆ…)β€˜suc 𝑏))
108 imaeq2 6055 . . . . . . . . . . . . 13 (𝑦 = 𝑐 β†’ ((rec(𝐺, βˆ…)β€˜suc 𝑏) β€œ 𝑦) = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β€œ 𝑐))
109108fveq2d 6895 . . . . . . . . . . . 12 (𝑦 = 𝑐 β†’ (πΉβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β€œ 𝑦)) = (πΉβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β€œ 𝑐)))
110 eqid 2732 . . . . . . . . . . . 12 (𝑦 ∈ 𝒫 dom (rec(𝐺, βˆ…)β€˜suc 𝑏) ↦ (πΉβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β€œ 𝑦))) = (𝑦 ∈ 𝒫 dom (rec(𝐺, βˆ…)β€˜suc 𝑏) ↦ (πΉβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β€œ 𝑦)))
111 fvex 6904 . . . . . . . . . . . 12 (πΉβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β€œ 𝑐)) ∈ V
112109, 110, 111fvmpt 6998 . . . . . . . . . . 11 (𝑐 ∈ 𝒫 dom (rec(𝐺, βˆ…)β€˜suc 𝑏) β†’ ((𝑦 ∈ 𝒫 dom (rec(𝐺, βˆ…)β€˜suc 𝑏) ↦ (πΉβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β€œ 𝑦)))β€˜π‘) = (πΉβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β€œ 𝑐)))
113107, 112syl 17 . . . . . . . . . 10 ((𝑏 ∈ Ο‰ ∧ 𝑐 ∈ (𝑅1β€˜suc 𝑏)) β†’ ((𝑦 ∈ 𝒫 dom (rec(𝐺, βˆ…)β€˜suc 𝑏) ↦ (πΉβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β€œ 𝑦)))β€˜π‘) = (πΉβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β€œ 𝑐)))
11497, 113eqtrid 2784 . . . . . . . . 9 ((𝑏 ∈ Ο‰ ∧ 𝑐 ∈ (𝑅1β€˜suc 𝑏)) β†’ ((πΊβ€˜(rec(𝐺, βˆ…)β€˜suc 𝑏))β€˜π‘) = (πΉβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β€œ 𝑐)))
11558, 86, 1143eqtr4d 2782 . . . . . . . 8 ((𝑏 ∈ Ο‰ ∧ 𝑐 ∈ (𝑅1β€˜suc 𝑏)) β†’ ((πΊβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)))β€˜π‘) = ((πΊβ€˜(rec(𝐺, βˆ…)β€˜suc 𝑏))β€˜π‘))
116115adantlr 713 . . . . . . 7 (((𝑏 ∈ Ο‰ ∧ (rec(𝐺, βˆ…)β€˜π‘) = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘))) ∧ 𝑐 ∈ (𝑅1β€˜suc 𝑏)) β†’ ((πΊβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)))β€˜π‘) = ((πΊβ€˜(rec(𝐺, βˆ…)β€˜suc 𝑏))β€˜π‘))
117 fveq2 6891 . . . . . . . . 9 ((rec(𝐺, βˆ…)β€˜π‘) = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β†’ (πΊβ€˜(rec(𝐺, βˆ…)β€˜π‘)) = (πΊβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘))))
118117fveq1d 6893 . . . . . . . 8 ((rec(𝐺, βˆ…)β€˜π‘) = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β†’ ((πΊβ€˜(rec(𝐺, βˆ…)β€˜π‘))β€˜π‘) = ((πΊβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)))β€˜π‘))
119118ad2antlr 725 . . . . . . 7 (((𝑏 ∈ Ο‰ ∧ (rec(𝐺, βˆ…)β€˜π‘) = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘))) ∧ 𝑐 ∈ (𝑅1β€˜suc 𝑏)) β†’ ((πΊβ€˜(rec(𝐺, βˆ…)β€˜π‘))β€˜π‘) = ((πΊβ€˜((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)))β€˜π‘))
120 rdgsuc 8426 . . . . . . . . . 10 (suc 𝑏 ∈ On β†’ (rec(𝐺, βˆ…)β€˜suc suc 𝑏) = (πΊβ€˜(rec(𝐺, βˆ…)β€˜suc 𝑏)))
12144, 120syl 17 . . . . . . . . 9 (𝑏 ∈ Ο‰ β†’ (rec(𝐺, βˆ…)β€˜suc suc 𝑏) = (πΊβ€˜(rec(𝐺, βˆ…)β€˜suc 𝑏)))
122121fveq1d 6893 . . . . . . . 8 (𝑏 ∈ Ο‰ β†’ ((rec(𝐺, βˆ…)β€˜suc suc 𝑏)β€˜π‘) = ((πΊβ€˜(rec(𝐺, βˆ…)β€˜suc 𝑏))β€˜π‘))
123122ad2antrr 724 . . . . . . 7 (((𝑏 ∈ Ο‰ ∧ (rec(𝐺, βˆ…)β€˜π‘) = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘))) ∧ 𝑐 ∈ (𝑅1β€˜suc 𝑏)) β†’ ((rec(𝐺, βˆ…)β€˜suc suc 𝑏)β€˜π‘) = ((πΊβ€˜(rec(𝐺, βˆ…)β€˜suc 𝑏))β€˜π‘))
124116, 119, 1233eqtr4rd 2783 . . . . . 6 (((𝑏 ∈ Ο‰ ∧ (rec(𝐺, βˆ…)β€˜π‘) = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘))) ∧ 𝑐 ∈ (𝑅1β€˜suc 𝑏)) β†’ ((rec(𝐺, βˆ…)β€˜suc suc 𝑏)β€˜π‘) = ((πΊβ€˜(rec(𝐺, βˆ…)β€˜π‘))β€˜π‘))
125 fvres 6910 . . . . . . 7 (𝑐 ∈ (𝑅1β€˜suc 𝑏) β†’ (((rec(𝐺, βˆ…)β€˜suc suc 𝑏) β†Ύ (𝑅1β€˜suc 𝑏))β€˜π‘) = ((rec(𝐺, βˆ…)β€˜suc suc 𝑏)β€˜π‘))
126125adantl 482 . . . . . 6 (((𝑏 ∈ Ο‰ ∧ (rec(𝐺, βˆ…)β€˜π‘) = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘))) ∧ 𝑐 ∈ (𝑅1β€˜suc 𝑏)) β†’ (((rec(𝐺, βˆ…)β€˜suc suc 𝑏) β†Ύ (𝑅1β€˜suc 𝑏))β€˜π‘) = ((rec(𝐺, βˆ…)β€˜suc suc 𝑏)β€˜π‘))
127 rdgsuc 8426 . . . . . . . . 9 (𝑏 ∈ On β†’ (rec(𝐺, βˆ…)β€˜suc 𝑏) = (πΊβ€˜(rec(𝐺, βˆ…)β€˜π‘)))
12850, 127syl 17 . . . . . . . 8 (𝑏 ∈ Ο‰ β†’ (rec(𝐺, βˆ…)β€˜suc 𝑏) = (πΊβ€˜(rec(𝐺, βˆ…)β€˜π‘)))
129128fveq1d 6893 . . . . . . 7 (𝑏 ∈ Ο‰ β†’ ((rec(𝐺, βˆ…)β€˜suc 𝑏)β€˜π‘) = ((πΊβ€˜(rec(𝐺, βˆ…)β€˜π‘))β€˜π‘))
130129ad2antrr 724 . . . . . 6 (((𝑏 ∈ Ο‰ ∧ (rec(𝐺, βˆ…)β€˜π‘) = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘))) ∧ 𝑐 ∈ (𝑅1β€˜suc 𝑏)) β†’ ((rec(𝐺, βˆ…)β€˜suc 𝑏)β€˜π‘) = ((πΊβ€˜(rec(𝐺, βˆ…)β€˜π‘))β€˜π‘))
131124, 126, 1303eqtr4rd 2783 . . . . 5 (((𝑏 ∈ Ο‰ ∧ (rec(𝐺, βˆ…)β€˜π‘) = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘))) ∧ 𝑐 ∈ (𝑅1β€˜suc 𝑏)) β†’ ((rec(𝐺, βˆ…)β€˜suc 𝑏)β€˜π‘) = (((rec(𝐺, βˆ…)β€˜suc suc 𝑏) β†Ύ (𝑅1β€˜suc 𝑏))β€˜π‘))
13238, 49, 131eqfnfvd 7035 . . . 4 ((𝑏 ∈ Ο‰ ∧ (rec(𝐺, βˆ…)β€˜π‘) = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘))) β†’ (rec(𝐺, βˆ…)β€˜suc 𝑏) = ((rec(𝐺, βˆ…)β€˜suc suc 𝑏) β†Ύ (𝑅1β€˜suc 𝑏)))
133132ex 413 . . 3 (𝑏 ∈ Ο‰ β†’ ((rec(𝐺, βˆ…)β€˜π‘) = ((rec(𝐺, βˆ…)β€˜suc 𝑏) β†Ύ (𝑅1β€˜π‘)) β†’ (rec(𝐺, βˆ…)β€˜suc 𝑏) = ((rec(𝐺, βˆ…)β€˜suc suc 𝑏) β†Ύ (𝑅1β€˜suc 𝑏))))
1346, 12, 18, 24, 30, 133finds 7891 . 2 (𝐴 ∈ Ο‰ β†’ (rec(𝐺, βˆ…)β€˜π΄) = ((rec(𝐺, βˆ…)β€˜suc 𝐴) β†Ύ (𝑅1β€˜π΄)))
135 resss 6006 . 2 ((rec(𝐺, βˆ…)β€˜suc 𝐴) β†Ύ (𝑅1β€˜π΄)) βŠ† (rec(𝐺, βˆ…)β€˜suc 𝐴)
136134, 135eqsstrdi 4036 1 (𝐴 ∈ Ο‰ β†’ (rec(𝐺, βˆ…)β€˜π΄) βŠ† (rec(𝐺, βˆ…)β€˜suc 𝐴))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602  {csn 4628  βˆͺ ciun 4997   ↦ cmpt 5231  Tr wtr 5265   Γ— cxp 5674  dom cdm 5676   β†Ύ cres 5678   β€œ cima 5679  Oncon0 6364  suc csuc 6366   Fn wfn 6538  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  Ο‰com 7857  reccrdg 8411  Fincfn 8941  π‘…1cr1 9759  cardccrd 9932
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  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-int 4951  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-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 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-r1 9761  df-dju 9898  df-card 9936
This theorem is referenced by:  ackbij2lem4  10239
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