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Theorem ackbij2lem3 9101
 Description: Lemma for ackbij2 9103. (Contributed by Stefan O'Rear, 18-Nov-2014.)
Hypotheses
Ref Expression
ackbij.f 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
ackbij.g 𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥𝑦))))
Assertion
Ref Expression
ackbij2lem3 (𝐴 ∈ ω → (rec(𝐺, ∅)‘𝐴) ⊆ (rec(𝐺, ∅)‘suc 𝐴))
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝐴,𝑦

Proof of Theorem ackbij2lem3
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6229 . . . 4 (𝑎 = ∅ → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘∅))
2 suceq 5828 . . . . . 6 (𝑎 = ∅ → suc 𝑎 = suc ∅)
32fveq2d 6233 . . . . 5 (𝑎 = ∅ → (rec(𝐺, ∅)‘suc 𝑎) = (rec(𝐺, ∅)‘suc ∅))
4 fveq2 6229 . . . . 5 (𝑎 = ∅ → (𝑅1𝑎) = (𝑅1‘∅))
53, 4reseq12d 5429 . . . 4 (𝑎 = ∅ → ((rec(𝐺, ∅)‘suc 𝑎) ↾ (𝑅1𝑎)) = ((rec(𝐺, ∅)‘suc ∅) ↾ (𝑅1‘∅)))
61, 5eqeq12d 2666 . . 3 (𝑎 = ∅ → ((rec(𝐺, ∅)‘𝑎) = ((rec(𝐺, ∅)‘suc 𝑎) ↾ (𝑅1𝑎)) ↔ (rec(𝐺, ∅)‘∅) = ((rec(𝐺, ∅)‘suc ∅) ↾ (𝑅1‘∅))))
7 fveq2 6229 . . . 4 (𝑎 = 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝑏))
8 suceq 5828 . . . . . 6 (𝑎 = 𝑏 → suc 𝑎 = suc 𝑏)
98fveq2d 6233 . . . . 5 (𝑎 = 𝑏 → (rec(𝐺, ∅)‘suc 𝑎) = (rec(𝐺, ∅)‘suc 𝑏))
10 fveq2 6229 . . . . 5 (𝑎 = 𝑏 → (𝑅1𝑎) = (𝑅1𝑏))
119, 10reseq12d 5429 . . . 4 (𝑎 = 𝑏 → ((rec(𝐺, ∅)‘suc 𝑎) ↾ (𝑅1𝑎)) = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)))
127, 11eqeq12d 2666 . . 3 (𝑎 = 𝑏 → ((rec(𝐺, ∅)‘𝑎) = ((rec(𝐺, ∅)‘suc 𝑎) ↾ (𝑅1𝑎)) ↔ (rec(𝐺, ∅)‘𝑏) = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏))))
13 fveq2 6229 . . . 4 (𝑎 = suc 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘suc 𝑏))
14 suceq 5828 . . . . . 6 (𝑎 = suc 𝑏 → suc 𝑎 = suc suc 𝑏)
1514fveq2d 6233 . . . . 5 (𝑎 = suc 𝑏 → (rec(𝐺, ∅)‘suc 𝑎) = (rec(𝐺, ∅)‘suc suc 𝑏))
16 fveq2 6229 . . . . 5 (𝑎 = suc 𝑏 → (𝑅1𝑎) = (𝑅1‘suc 𝑏))
1715, 16reseq12d 5429 . . . 4 (𝑎 = suc 𝑏 → ((rec(𝐺, ∅)‘suc 𝑎) ↾ (𝑅1𝑎)) = ((rec(𝐺, ∅)‘suc suc 𝑏) ↾ (𝑅1‘suc 𝑏)))
1813, 17eqeq12d 2666 . . 3 (𝑎 = suc 𝑏 → ((rec(𝐺, ∅)‘𝑎) = ((rec(𝐺, ∅)‘suc 𝑎) ↾ (𝑅1𝑎)) ↔ (rec(𝐺, ∅)‘suc 𝑏) = ((rec(𝐺, ∅)‘suc suc 𝑏) ↾ (𝑅1‘suc 𝑏))))
19 fveq2 6229 . . . 4 (𝑎 = 𝐴 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝐴))
20 suceq 5828 . . . . . 6 (𝑎 = 𝐴 → suc 𝑎 = suc 𝐴)
2120fveq2d 6233 . . . . 5 (𝑎 = 𝐴 → (rec(𝐺, ∅)‘suc 𝑎) = (rec(𝐺, ∅)‘suc 𝐴))
22 fveq2 6229 . . . . 5 (𝑎 = 𝐴 → (𝑅1𝑎) = (𝑅1𝐴))
2321, 22reseq12d 5429 . . . 4 (𝑎 = 𝐴 → ((rec(𝐺, ∅)‘suc 𝑎) ↾ (𝑅1𝑎)) = ((rec(𝐺, ∅)‘suc 𝐴) ↾ (𝑅1𝐴)))
2419, 23eqeq12d 2666 . . 3 (𝑎 = 𝐴 → ((rec(𝐺, ∅)‘𝑎) = ((rec(𝐺, ∅)‘suc 𝑎) ↾ (𝑅1𝑎)) ↔ (rec(𝐺, ∅)‘𝐴) = ((rec(𝐺, ∅)‘suc 𝐴) ↾ (𝑅1𝐴))))
25 res0 5432 . . . 4 ((rec(𝐺, ∅)‘suc ∅) ↾ ∅) = ∅
26 r10 8669 . . . . 5 (𝑅1‘∅) = ∅
2726reseq2i 5425 . . . 4 ((rec(𝐺, ∅)‘suc ∅) ↾ (𝑅1‘∅)) = ((rec(𝐺, ∅)‘suc ∅) ↾ ∅)
28 0ex 4823 . . . . 5 ∅ ∈ V
2928rdg0 7562 . . . 4 (rec(𝐺, ∅)‘∅) = ∅
3025, 27, 293eqtr4ri 2684 . . 3 (rec(𝐺, ∅)‘∅) = ((rec(𝐺, ∅)‘suc ∅) ↾ (𝑅1‘∅))
31 peano2 7128 . . . . . . . 8 (𝑏 ∈ ω → suc 𝑏 ∈ ω)
32 ackbij.f . . . . . . . . 9 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
33 ackbij.g . . . . . . . . 9 𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥𝑦))))
3432, 33ackbij2lem2 9100 . . . . . . . 8 (suc 𝑏 ∈ ω → (rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)))
3531, 34syl 17 . . . . . . 7 (𝑏 ∈ ω → (rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)))
36 f1ofn 6176 . . . . . . 7 ((rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)) → (rec(𝐺, ∅)‘suc 𝑏) Fn (𝑅1‘suc 𝑏))
3735, 36syl 17 . . . . . 6 (𝑏 ∈ ω → (rec(𝐺, ∅)‘suc 𝑏) Fn (𝑅1‘suc 𝑏))
3837adantr 480 . . . . 5 ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏) = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏))) → (rec(𝐺, ∅)‘suc 𝑏) Fn (𝑅1‘suc 𝑏))
39 peano2 7128 . . . . . . . 8 (suc 𝑏 ∈ ω → suc suc 𝑏 ∈ ω)
4032, 33ackbij2lem2 9100 . . . . . . . 8 (suc suc 𝑏 ∈ ω → (rec(𝐺, ∅)‘suc suc 𝑏):(𝑅1‘suc suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc suc 𝑏)))
41 f1ofn 6176 . . . . . . . 8 ((rec(𝐺, ∅)‘suc suc 𝑏):(𝑅1‘suc suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc suc 𝑏)) → (rec(𝐺, ∅)‘suc suc 𝑏) Fn (𝑅1‘suc suc 𝑏))
4231, 39, 40, 414syl 19 . . . . . . 7 (𝑏 ∈ ω → (rec(𝐺, ∅)‘suc suc 𝑏) Fn (𝑅1‘suc suc 𝑏))
43 nnon 7113 . . . . . . . . 9 (suc 𝑏 ∈ ω → suc 𝑏 ∈ On)
4431, 43syl 17 . . . . . . . 8 (𝑏 ∈ ω → suc 𝑏 ∈ On)
45 r1sssuc 8684 . . . . . . . 8 (suc 𝑏 ∈ On → (𝑅1‘suc 𝑏) ⊆ (𝑅1‘suc suc 𝑏))
4644, 45syl 17 . . . . . . 7 (𝑏 ∈ ω → (𝑅1‘suc 𝑏) ⊆ (𝑅1‘suc suc 𝑏))
47 fnssres 6042 . . . . . . 7 (((rec(𝐺, ∅)‘suc suc 𝑏) Fn (𝑅1‘suc suc 𝑏) ∧ (𝑅1‘suc 𝑏) ⊆ (𝑅1‘suc suc 𝑏)) → ((rec(𝐺, ∅)‘suc suc 𝑏) ↾ (𝑅1‘suc 𝑏)) Fn (𝑅1‘suc 𝑏))
4842, 46, 47syl2anc 694 . . . . . 6 (𝑏 ∈ ω → ((rec(𝐺, ∅)‘suc suc 𝑏) ↾ (𝑅1‘suc 𝑏)) Fn (𝑅1‘suc 𝑏))
4948adantr 480 . . . . 5 ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏) = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏))) → ((rec(𝐺, ∅)‘suc suc 𝑏) ↾ (𝑅1‘suc 𝑏)) Fn (𝑅1‘suc 𝑏))
50 nnon 7113 . . . . . . . . . . . . . . 15 (𝑏 ∈ ω → 𝑏 ∈ On)
51 r1suc 8671 . . . . . . . . . . . . . . 15 (𝑏 ∈ On → (𝑅1‘suc 𝑏) = 𝒫 (𝑅1𝑏))
5250, 51syl 17 . . . . . . . . . . . . . 14 (𝑏 ∈ ω → (𝑅1‘suc 𝑏) = 𝒫 (𝑅1𝑏))
5352eleq2d 2716 . . . . . . . . . . . . 13 (𝑏 ∈ ω → (𝑐 ∈ (𝑅1‘suc 𝑏) ↔ 𝑐 ∈ 𝒫 (𝑅1𝑏)))
5453biimpa 500 . . . . . . . . . . . 12 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → 𝑐 ∈ 𝒫 (𝑅1𝑏))
5554elpwid 4203 . . . . . . . . . . 11 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → 𝑐 ⊆ (𝑅1𝑏))
56 resima2 5467 . . . . . . . . . . 11 (𝑐 ⊆ (𝑅1𝑏) → (((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑐) = ((rec(𝐺, ∅)‘suc 𝑏) “ 𝑐))
5755, 56syl 17 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → (((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑐) = ((rec(𝐺, ∅)‘suc 𝑏) “ 𝑐))
5857fveq2d 6233 . . . . . . . . 9 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑐)) = (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑐)))
59 fvex 6239 . . . . . . . . . . . . 13 (rec(𝐺, ∅)‘suc 𝑏) ∈ V
6059resex 5478 . . . . . . . . . . . 12 ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) ∈ V
61 dmeq 5356 . . . . . . . . . . . . . . 15 (𝑥 = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) → dom 𝑥 = dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)))
6261pweqd 4196 . . . . . . . . . . . . . 14 (𝑥 = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) → 𝒫 dom 𝑥 = 𝒫 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)))
63 imaeq1 5496 . . . . . . . . . . . . . . 15 (𝑥 = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) → (𝑥𝑦) = (((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑦))
6463fveq2d 6233 . . . . . . . . . . . . . 14 (𝑥 = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) → (𝐹‘(𝑥𝑦)) = (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑦)))
6562, 64mpteq12dv 4766 . . . . . . . . . . . . 13 (𝑥 = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) → (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥𝑦))) = (𝑦 ∈ 𝒫 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) ↦ (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑦))))
6660dmex 7141 . . . . . . . . . . . . . . 15 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) ∈ V
6766pwex 4878 . . . . . . . . . . . . . 14 𝒫 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) ∈ V
6867mptex 6527 . . . . . . . . . . . . 13 (𝑦 ∈ 𝒫 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) ↦ (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑦))) ∈ V
6965, 33, 68fvmpt 6321 . . . . . . . . . . . 12 (((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) ∈ V → (𝐺‘((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏))) = (𝑦 ∈ 𝒫 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) ↦ (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑦))))
7060, 69ax-mp 5 . . . . . . . . . . 11 (𝐺‘((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏))) = (𝑦 ∈ 𝒫 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) ↦ (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑦)))
7170fveq1i 6230 . . . . . . . . . 10 ((𝐺‘((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)))‘𝑐) = ((𝑦 ∈ 𝒫 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) ↦ (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑦)))‘𝑐)
72 r1sssuc 8684 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ On → (𝑅1𝑏) ⊆ (𝑅1‘suc 𝑏))
7350, 72syl 17 . . . . . . . . . . . . . . . 16 (𝑏 ∈ ω → (𝑅1𝑏) ⊆ (𝑅1‘suc 𝑏))
74 fnssres 6042 . . . . . . . . . . . . . . . 16 (((rec(𝐺, ∅)‘suc 𝑏) Fn (𝑅1‘suc 𝑏) ∧ (𝑅1𝑏) ⊆ (𝑅1‘suc 𝑏)) → ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) Fn (𝑅1𝑏))
7537, 73, 74syl2anc 694 . . . . . . . . . . . . . . 15 (𝑏 ∈ ω → ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) Fn (𝑅1𝑏))
76 fndm 6028 . . . . . . . . . . . . . . 15 (((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) Fn (𝑅1𝑏) → dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) = (𝑅1𝑏))
7775, 76syl 17 . . . . . . . . . . . . . 14 (𝑏 ∈ ω → dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) = (𝑅1𝑏))
7877pweqd 4196 . . . . . . . . . . . . 13 (𝑏 ∈ ω → 𝒫 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) = 𝒫 (𝑅1𝑏))
7978adantr 480 . . . . . . . . . . . 12 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → 𝒫 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) = 𝒫 (𝑅1𝑏))
8054, 79eleqtrrd 2733 . . . . . . . . . . 11 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → 𝑐 ∈ 𝒫 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)))
81 imaeq2 5497 . . . . . . . . . . . . 13 (𝑦 = 𝑐 → (((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑦) = (((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑐))
8281fveq2d 6233 . . . . . . . . . . . 12 (𝑦 = 𝑐 → (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑦)) = (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑐)))
83 eqid 2651 . . . . . . . . . . . 12 (𝑦 ∈ 𝒫 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) ↦ (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑦))) = (𝑦 ∈ 𝒫 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) ↦ (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑦)))
84 fvex 6239 . . . . . . . . . . . 12 (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑐)) ∈ V
8582, 83, 84fvmpt 6321 . . . . . . . . . . 11 (𝑐 ∈ 𝒫 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) → ((𝑦 ∈ 𝒫 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) ↦ (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑦)))‘𝑐) = (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑐)))
8680, 85syl 17 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → ((𝑦 ∈ 𝒫 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) ↦ (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑦)))‘𝑐) = (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑐)))
8771, 86syl5eq 2697 . . . . . . . . 9 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → ((𝐺‘((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)))‘𝑐) = (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑐)))
88 dmeq 5356 . . . . . . . . . . . . . . 15 (𝑥 = (rec(𝐺, ∅)‘suc 𝑏) → dom 𝑥 = dom (rec(𝐺, ∅)‘suc 𝑏))
8988pweqd 4196 . . . . . . . . . . . . . 14 (𝑥 = (rec(𝐺, ∅)‘suc 𝑏) → 𝒫 dom 𝑥 = 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏))
90 imaeq1 5496 . . . . . . . . . . . . . . 15 (𝑥 = (rec(𝐺, ∅)‘suc 𝑏) → (𝑥𝑦) = ((rec(𝐺, ∅)‘suc 𝑏) “ 𝑦))
9190fveq2d 6233 . . . . . . . . . . . . . 14 (𝑥 = (rec(𝐺, ∅)‘suc 𝑏) → (𝐹‘(𝑥𝑦)) = (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑦)))
9289, 91mpteq12dv 4766 . . . . . . . . . . . . 13 (𝑥 = (rec(𝐺, ∅)‘suc 𝑏) → (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥𝑦))) = (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑦))))
9359dmex 7141 . . . . . . . . . . . . . . 15 dom (rec(𝐺, ∅)‘suc 𝑏) ∈ V
9493pwex 4878 . . . . . . . . . . . . . 14 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏) ∈ V
9594mptex 6527 . . . . . . . . . . . . 13 (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑦))) ∈ V
9692, 33, 95fvmpt 6321 . . . . . . . . . . . 12 ((rec(𝐺, ∅)‘suc 𝑏) ∈ V → (𝐺‘(rec(𝐺, ∅)‘suc 𝑏)) = (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑦))))
9759, 96ax-mp 5 . . . . . . . . . . 11 (𝐺‘(rec(𝐺, ∅)‘suc 𝑏)) = (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑦)))
9897fveq1i 6230 . . . . . . . . . 10 ((𝐺‘(rec(𝐺, ∅)‘suc 𝑏))‘𝑐) = ((𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑦)))‘𝑐)
99 r1tr 8677 . . . . . . . . . . . . . . 15 Tr (𝑅1‘suc 𝑏)
10099a1i 11 . . . . . . . . . . . . . 14 (𝑏 ∈ ω → Tr (𝑅1‘suc 𝑏))
101 dftr4 4790 . . . . . . . . . . . . . 14 (Tr (𝑅1‘suc 𝑏) ↔ (𝑅1‘suc 𝑏) ⊆ 𝒫 (𝑅1‘suc 𝑏))
102100, 101sylib 208 . . . . . . . . . . . . 13 (𝑏 ∈ ω → (𝑅1‘suc 𝑏) ⊆ 𝒫 (𝑅1‘suc 𝑏))
103102sselda 3636 . . . . . . . . . . . 12 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → 𝑐 ∈ 𝒫 (𝑅1‘suc 𝑏))
104 f1odm 6179 . . . . . . . . . . . . . . 15 ((rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)) → dom (rec(𝐺, ∅)‘suc 𝑏) = (𝑅1‘suc 𝑏))
10535, 104syl 17 . . . . . . . . . . . . . 14 (𝑏 ∈ ω → dom (rec(𝐺, ∅)‘suc 𝑏) = (𝑅1‘suc 𝑏))
106105pweqd 4196 . . . . . . . . . . . . 13 (𝑏 ∈ ω → 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏) = 𝒫 (𝑅1‘suc 𝑏))
107106adantr 480 . . . . . . . . . . . 12 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏) = 𝒫 (𝑅1‘suc 𝑏))
108103, 107eleqtrrd 2733 . . . . . . . . . . 11 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → 𝑐 ∈ 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏))
109 imaeq2 5497 . . . . . . . . . . . . 13 (𝑦 = 𝑐 → ((rec(𝐺, ∅)‘suc 𝑏) “ 𝑦) = ((rec(𝐺, ∅)‘suc 𝑏) “ 𝑐))
110109fveq2d 6233 . . . . . . . . . . . 12 (𝑦 = 𝑐 → (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑦)) = (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑐)))
111 eqid 2651 . . . . . . . . . . . 12 (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑦))) = (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑦)))
112 fvex 6239 . . . . . . . . . . . 12 (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑐)) ∈ V
113110, 111, 112fvmpt 6321 . . . . . . . . . . 11 (𝑐 ∈ 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏) → ((𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑦)))‘𝑐) = (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑐)))
114108, 113syl 17 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → ((𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑦)))‘𝑐) = (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑐)))
11598, 114syl5eq 2697 . . . . . . . . 9 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → ((𝐺‘(rec(𝐺, ∅)‘suc 𝑏))‘𝑐) = (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑐)))
11658, 87, 1153eqtr4d 2695 . . . . . . . 8 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → ((𝐺‘((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)))‘𝑐) = ((𝐺‘(rec(𝐺, ∅)‘suc 𝑏))‘𝑐))
117116adantlr 751 . . . . . . 7 (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏) = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏))) ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → ((𝐺‘((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)))‘𝑐) = ((𝐺‘(rec(𝐺, ∅)‘suc 𝑏))‘𝑐))
118 fveq2 6229 . . . . . . . . 9 ((rec(𝐺, ∅)‘𝑏) = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) → (𝐺‘(rec(𝐺, ∅)‘𝑏)) = (𝐺‘((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏))))
119118fveq1d 6231 . . . . . . . 8 ((rec(𝐺, ∅)‘𝑏) = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) → ((𝐺‘(rec(𝐺, ∅)‘𝑏))‘𝑐) = ((𝐺‘((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)))‘𝑐))
120119ad2antlr 763 . . . . . . 7 (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏) = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏))) ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → ((𝐺‘(rec(𝐺, ∅)‘𝑏))‘𝑐) = ((𝐺‘((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)))‘𝑐))
121 rdgsuc 7565 . . . . . . . . . 10 (suc 𝑏 ∈ On → (rec(𝐺, ∅)‘suc suc 𝑏) = (𝐺‘(rec(𝐺, ∅)‘suc 𝑏)))
12244, 121syl 17 . . . . . . . . 9 (𝑏 ∈ ω → (rec(𝐺, ∅)‘suc suc 𝑏) = (𝐺‘(rec(𝐺, ∅)‘suc 𝑏)))
123122fveq1d 6231 . . . . . . . 8 (𝑏 ∈ ω → ((rec(𝐺, ∅)‘suc suc 𝑏)‘𝑐) = ((𝐺‘(rec(𝐺, ∅)‘suc 𝑏))‘𝑐))
124123ad2antrr 762 . . . . . . 7 (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏) = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏))) ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → ((rec(𝐺, ∅)‘suc suc 𝑏)‘𝑐) = ((𝐺‘(rec(𝐺, ∅)‘suc 𝑏))‘𝑐))
125117, 120, 1243eqtr4rd 2696 . . . . . 6 (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏) = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏))) ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → ((rec(𝐺, ∅)‘suc suc 𝑏)‘𝑐) = ((𝐺‘(rec(𝐺, ∅)‘𝑏))‘𝑐))
126 fvres 6245 . . . . . . 7 (𝑐 ∈ (𝑅1‘suc 𝑏) → (((rec(𝐺, ∅)‘suc suc 𝑏) ↾ (𝑅1‘suc 𝑏))‘𝑐) = ((rec(𝐺, ∅)‘suc suc 𝑏)‘𝑐))
127126adantl 481 . . . . . 6 (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏) = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏))) ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → (((rec(𝐺, ∅)‘suc suc 𝑏) ↾ (𝑅1‘suc 𝑏))‘𝑐) = ((rec(𝐺, ∅)‘suc suc 𝑏)‘𝑐))
128 rdgsuc 7565 . . . . . . . . 9 (𝑏 ∈ On → (rec(𝐺, ∅)‘suc 𝑏) = (𝐺‘(rec(𝐺, ∅)‘𝑏)))
12950, 128syl 17 . . . . . . . 8 (𝑏 ∈ ω → (rec(𝐺, ∅)‘suc 𝑏) = (𝐺‘(rec(𝐺, ∅)‘𝑏)))
130129fveq1d 6231 . . . . . . 7 (𝑏 ∈ ω → ((rec(𝐺, ∅)‘suc 𝑏)‘𝑐) = ((𝐺‘(rec(𝐺, ∅)‘𝑏))‘𝑐))
131130ad2antrr 762 . . . . . 6 (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏) = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏))) ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → ((rec(𝐺, ∅)‘suc 𝑏)‘𝑐) = ((𝐺‘(rec(𝐺, ∅)‘𝑏))‘𝑐))
132125, 127, 1313eqtr4rd 2696 . . . . 5 (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏) = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏))) ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → ((rec(𝐺, ∅)‘suc 𝑏)‘𝑐) = (((rec(𝐺, ∅)‘suc suc 𝑏) ↾ (𝑅1‘suc 𝑏))‘𝑐))
13338, 49, 132eqfnfvd 6354 . . . 4 ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏) = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏))) → (rec(𝐺, ∅)‘suc 𝑏) = ((rec(𝐺, ∅)‘suc suc 𝑏) ↾ (𝑅1‘suc 𝑏)))
134133ex 449 . . 3 (𝑏 ∈ ω → ((rec(𝐺, ∅)‘𝑏) = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) → (rec(𝐺, ∅)‘suc 𝑏) = ((rec(𝐺, ∅)‘suc suc 𝑏) ↾ (𝑅1‘suc 𝑏))))
1356, 12, 18, 24, 30, 134finds 7134 . 2 (𝐴 ∈ ω → (rec(𝐺, ∅)‘𝐴) = ((rec(𝐺, ∅)‘suc 𝐴) ↾ (𝑅1𝐴)))
136 resss 5457 . 2 ((rec(𝐺, ∅)‘suc 𝐴) ↾ (𝑅1𝐴)) ⊆ (rec(𝐺, ∅)‘suc 𝐴)
137135, 136syl6eqss 3688 1 (𝐴 ∈ ω → (rec(𝐺, ∅)‘𝐴) ⊆ (rec(𝐺, ∅)‘suc 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  𝒫 cpw 4191  {csn 4210  ∪ ciun 4552   ↦ cmpt 4762  Tr wtr 4785   × cxp 5141  dom cdm 5143   ↾ cres 5145   “ cima 5146  Oncon0 5761  suc csuc 5763   Fn wfn 5921  –1-1-onto→wf1o 5925  ‘cfv 5926  ωcom 7107  reccrdg 7550  Fincfn 7997  𝑅1cr1 8663  cardccrd 8799 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-r1 8665  df-card 8803  df-cda 9028 This theorem is referenced by:  ackbij2lem4  9102
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