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Theorem ackbij2lem3 10177
Description: Lemma for ackbij2 10179. (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 6842 . . . 4 (𝑎 = ∅ → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘∅))
2 suceq 6383 . . . . . 6 (𝑎 = ∅ → suc 𝑎 = suc ∅)
32fveq2d 6846 . . . . 5 (𝑎 = ∅ → (rec(𝐺, ∅)‘suc 𝑎) = (rec(𝐺, ∅)‘suc ∅))
4 fveq2 6842 . . . . 5 (𝑎 = ∅ → (𝑅1𝑎) = (𝑅1‘∅))
53, 4reseq12d 5938 . . . 4 (𝑎 = ∅ → ((rec(𝐺, ∅)‘suc 𝑎) ↾ (𝑅1𝑎)) = ((rec(𝐺, ∅)‘suc ∅) ↾ (𝑅1‘∅)))
61, 5eqeq12d 2752 . . 3 (𝑎 = ∅ → ((rec(𝐺, ∅)‘𝑎) = ((rec(𝐺, ∅)‘suc 𝑎) ↾ (𝑅1𝑎)) ↔ (rec(𝐺, ∅)‘∅) = ((rec(𝐺, ∅)‘suc ∅) ↾ (𝑅1‘∅))))
7 fveq2 6842 . . . 4 (𝑎 = 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝑏))
8 suceq 6383 . . . . . 6 (𝑎 = 𝑏 → suc 𝑎 = suc 𝑏)
98fveq2d 6846 . . . . 5 (𝑎 = 𝑏 → (rec(𝐺, ∅)‘suc 𝑎) = (rec(𝐺, ∅)‘suc 𝑏))
10 fveq2 6842 . . . . 5 (𝑎 = 𝑏 → (𝑅1𝑎) = (𝑅1𝑏))
119, 10reseq12d 5938 . . . 4 (𝑎 = 𝑏 → ((rec(𝐺, ∅)‘suc 𝑎) ↾ (𝑅1𝑎)) = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)))
127, 11eqeq12d 2752 . . 3 (𝑎 = 𝑏 → ((rec(𝐺, ∅)‘𝑎) = ((rec(𝐺, ∅)‘suc 𝑎) ↾ (𝑅1𝑎)) ↔ (rec(𝐺, ∅)‘𝑏) = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏))))
13 fveq2 6842 . . . 4 (𝑎 = suc 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘suc 𝑏))
14 suceq 6383 . . . . . 6 (𝑎 = suc 𝑏 → suc 𝑎 = suc suc 𝑏)
1514fveq2d 6846 . . . . 5 (𝑎 = suc 𝑏 → (rec(𝐺, ∅)‘suc 𝑎) = (rec(𝐺, ∅)‘suc suc 𝑏))
16 fveq2 6842 . . . . 5 (𝑎 = suc 𝑏 → (𝑅1𝑎) = (𝑅1‘suc 𝑏))
1715, 16reseq12d 5938 . . . 4 (𝑎 = suc 𝑏 → ((rec(𝐺, ∅)‘suc 𝑎) ↾ (𝑅1𝑎)) = ((rec(𝐺, ∅)‘suc suc 𝑏) ↾ (𝑅1‘suc 𝑏)))
1813, 17eqeq12d 2752 . . 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 6842 . . . 4 (𝑎 = 𝐴 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝐴))
20 suceq 6383 . . . . . 6 (𝑎 = 𝐴 → suc 𝑎 = suc 𝐴)
2120fveq2d 6846 . . . . 5 (𝑎 = 𝐴 → (rec(𝐺, ∅)‘suc 𝑎) = (rec(𝐺, ∅)‘suc 𝐴))
22 fveq2 6842 . . . . 5 (𝑎 = 𝐴 → (𝑅1𝑎) = (𝑅1𝐴))
2321, 22reseq12d 5938 . . . 4 (𝑎 = 𝐴 → ((rec(𝐺, ∅)‘suc 𝑎) ↾ (𝑅1𝑎)) = ((rec(𝐺, ∅)‘suc 𝐴) ↾ (𝑅1𝐴)))
2419, 23eqeq12d 2752 . . 3 (𝑎 = 𝐴 → ((rec(𝐺, ∅)‘𝑎) = ((rec(𝐺, ∅)‘suc 𝑎) ↾ (𝑅1𝑎)) ↔ (rec(𝐺, ∅)‘𝐴) = ((rec(𝐺, ∅)‘suc 𝐴) ↾ (𝑅1𝐴))))
25 res0 5941 . . . 4 ((rec(𝐺, ∅)‘suc ∅) ↾ ∅) = ∅
26 r10 9704 . . . . 5 (𝑅1‘∅) = ∅
2726reseq2i 5934 . . . 4 ((rec(𝐺, ∅)‘suc ∅) ↾ (𝑅1‘∅)) = ((rec(𝐺, ∅)‘suc ∅) ↾ ∅)
28 0ex 5264 . . . . 5 ∅ ∈ V
2928rdg0 8367 . . . 4 (rec(𝐺, ∅)‘∅) = ∅
3025, 27, 293eqtr4ri 2775 . . 3 (rec(𝐺, ∅)‘∅) = ((rec(𝐺, ∅)‘suc ∅) ↾ (𝑅1‘∅))
31 peano2 7827 . . . . . . . 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 10176 . . . . . . . 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 6785 . . . . . . 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 481 . . . . 5 ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏) = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏))) → (rec(𝐺, ∅)‘suc 𝑏) Fn (𝑅1‘suc 𝑏))
39 peano2 7827 . . . . . . . 8 (suc 𝑏 ∈ ω → suc suc 𝑏 ∈ ω)
4032, 33ackbij2lem2 10176 . . . . . . . 8 (suc suc 𝑏 ∈ ω → (rec(𝐺, ∅)‘suc suc 𝑏):(𝑅1‘suc suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc suc 𝑏)))
41 f1ofn 6785 . . . . . . . 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 7808 . . . . . . . . 9 (suc 𝑏 ∈ ω → suc 𝑏 ∈ On)
4431, 43syl 17 . . . . . . . 8 (𝑏 ∈ ω → suc 𝑏 ∈ On)
45 r1sssuc 9719 . . . . . . . 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 6624 . . . . . . 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 584 . . . . . 6 (𝑏 ∈ ω → ((rec(𝐺, ∅)‘suc suc 𝑏) ↾ (𝑅1‘suc 𝑏)) Fn (𝑅1‘suc 𝑏))
4948adantr 481 . . . . 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 7808 . . . . . . . . . . . . . . 15 (𝑏 ∈ ω → 𝑏 ∈ On)
51 r1suc 9706 . . . . . . . . . . . . . . 15 (𝑏 ∈ On → (𝑅1‘suc 𝑏) = 𝒫 (𝑅1𝑏))
5250, 51syl 17 . . . . . . . . . . . . . 14 (𝑏 ∈ ω → (𝑅1‘suc 𝑏) = 𝒫 (𝑅1𝑏))
5352eleq2d 2823 . . . . . . . . . . . . 13 (𝑏 ∈ ω → (𝑐 ∈ (𝑅1‘suc 𝑏) ↔ 𝑐 ∈ 𝒫 (𝑅1𝑏)))
5453biimpa 477 . . . . . . . . . . . 12 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → 𝑐 ∈ 𝒫 (𝑅1𝑏))
5554elpwid 4569 . . . . . . . . . . 11 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → 𝑐 ⊆ (𝑅1𝑏))
56 resima2 5972 . . . . . . . . . . 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 6846 . . . . . . . . 9 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑐)) = (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑐)))
59 fvex 6855 . . . . . . . . . . . . 13 (rec(𝐺, ∅)‘suc 𝑏) ∈ V
6059resex 5985 . . . . . . . . . . . 12 ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) ∈ V
61 dmeq 5859 . . . . . . . . . . . . . . 15 (𝑥 = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) → dom 𝑥 = dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)))
6261pweqd 4577 . . . . . . . . . . . . . 14 (𝑥 = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) → 𝒫 dom 𝑥 = 𝒫 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)))
63 imaeq1 6008 . . . . . . . . . . . . . . 15 (𝑥 = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) → (𝑥𝑦) = (((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑦))
6463fveq2d 6846 . . . . . . . . . . . . . 14 (𝑥 = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) → (𝐹‘(𝑥𝑦)) = (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑦)))
6562, 64mpteq12dv 5196 . . . . . . . . . . . . 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 7848 . . . . . . . . . . . . . . 15 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) ∈ V
6766pwex 5335 . . . . . . . . . . . . . 14 𝒫 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) ∈ V
6867mptex 7173 . . . . . . . . . . . . 13 (𝑦 ∈ 𝒫 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) ↦ (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑦))) ∈ V
6965, 33, 68fvmpt 6948 . . . . . . . . . . . 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 6843 . . . . . . . . . 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 9719 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ On → (𝑅1𝑏) ⊆ (𝑅1‘suc 𝑏))
7350, 72syl 17 . . . . . . . . . . . . . . . 16 (𝑏 ∈ ω → (𝑅1𝑏) ⊆ (𝑅1‘suc 𝑏))
74 fnssres 6624 . . . . . . . . . . . . . . . 16 (((rec(𝐺, ∅)‘suc 𝑏) Fn (𝑅1‘suc 𝑏) ∧ (𝑅1𝑏) ⊆ (𝑅1‘suc 𝑏)) → ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) Fn (𝑅1𝑏))
7537, 73, 74syl2anc 584 . . . . . . . . . . . . . . 15 (𝑏 ∈ ω → ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) Fn (𝑅1𝑏))
7675fndmd 6607 . . . . . . . . . . . . . 14 (𝑏 ∈ ω → dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) = (𝑅1𝑏))
7776pweqd 4577 . . . . . . . . . . . . 13 (𝑏 ∈ ω → 𝒫 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) = 𝒫 (𝑅1𝑏))
7877adantr 481 . . . . . . . . . . . 12 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → 𝒫 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) = 𝒫 (𝑅1𝑏))
7954, 78eleqtrrd 2841 . . . . . . . . . . 11 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → 𝑐 ∈ 𝒫 dom ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)))
80 imaeq2 6009 . . . . . . . . . . . . 13 (𝑦 = 𝑐 → (((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑦) = (((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑐))
8180fveq2d 6846 . . . . . . . . . . . 12 (𝑦 = 𝑐 → (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑦)) = (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑐)))
82 eqid 2736 . . . . . . . . . . . 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)))
83 fvex 6855 . . . . . . . . . . . 12 (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑐)) ∈ V
8481, 82, 83fvmpt 6948 . . . . . . . . . . 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)))
8579, 84syl 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)))
8671, 85eqtrid 2788 . . . . . . . . 9 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → ((𝐺‘((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)))‘𝑐) = (𝐹‘(((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) “ 𝑐)))
87 dmeq 5859 . . . . . . . . . . . . . . 15 (𝑥 = (rec(𝐺, ∅)‘suc 𝑏) → dom 𝑥 = dom (rec(𝐺, ∅)‘suc 𝑏))
8887pweqd 4577 . . . . . . . . . . . . . 14 (𝑥 = (rec(𝐺, ∅)‘suc 𝑏) → 𝒫 dom 𝑥 = 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏))
89 imaeq1 6008 . . . . . . . . . . . . . . 15 (𝑥 = (rec(𝐺, ∅)‘suc 𝑏) → (𝑥𝑦) = ((rec(𝐺, ∅)‘suc 𝑏) “ 𝑦))
9089fveq2d 6846 . . . . . . . . . . . . . 14 (𝑥 = (rec(𝐺, ∅)‘suc 𝑏) → (𝐹‘(𝑥𝑦)) = (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑦)))
9188, 90mpteq12dv 5196 . . . . . . . . . . . . 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))))
9259dmex 7848 . . . . . . . . . . . . . . 15 dom (rec(𝐺, ∅)‘suc 𝑏) ∈ V
9392pwex 5335 . . . . . . . . . . . . . 14 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏) ∈ V
9493mptex 7173 . . . . . . . . . . . . 13 (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑦))) ∈ V
9591, 33, 94fvmpt 6948 . . . . . . . . . . . 12 ((rec(𝐺, ∅)‘suc 𝑏) ∈ V → (𝐺‘(rec(𝐺, ∅)‘suc 𝑏)) = (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑦))))
9659, 95ax-mp 5 . . . . . . . . . . 11 (𝐺‘(rec(𝐺, ∅)‘suc 𝑏)) = (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑦)))
9796fveq1i 6843 . . . . . . . . . 10 ((𝐺‘(rec(𝐺, ∅)‘suc 𝑏))‘𝑐) = ((𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑦)))‘𝑐)
98 r1tr 9712 . . . . . . . . . . . . . . 15 Tr (𝑅1‘suc 𝑏)
9998a1i 11 . . . . . . . . . . . . . 14 (𝑏 ∈ ω → Tr (𝑅1‘suc 𝑏))
100 dftr4 5229 . . . . . . . . . . . . . 14 (Tr (𝑅1‘suc 𝑏) ↔ (𝑅1‘suc 𝑏) ⊆ 𝒫 (𝑅1‘suc 𝑏))
10199, 100sylib 217 . . . . . . . . . . . . 13 (𝑏 ∈ ω → (𝑅1‘suc 𝑏) ⊆ 𝒫 (𝑅1‘suc 𝑏))
102101sselda 3944 . . . . . . . . . . . 12 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → 𝑐 ∈ 𝒫 (𝑅1‘suc 𝑏))
103 f1odm 6788 . . . . . . . . . . . . . . 15 ((rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)) → dom (rec(𝐺, ∅)‘suc 𝑏) = (𝑅1‘suc 𝑏))
10435, 103syl 17 . . . . . . . . . . . . . 14 (𝑏 ∈ ω → dom (rec(𝐺, ∅)‘suc 𝑏) = (𝑅1‘suc 𝑏))
105104pweqd 4577 . . . . . . . . . . . . 13 (𝑏 ∈ ω → 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏) = 𝒫 (𝑅1‘suc 𝑏))
106105adantr 481 . . . . . . . . . . . 12 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏) = 𝒫 (𝑅1‘suc 𝑏))
107102, 106eleqtrrd 2841 . . . . . . . . . . 11 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → 𝑐 ∈ 𝒫 dom (rec(𝐺, ∅)‘suc 𝑏))
108 imaeq2 6009 . . . . . . . . . . . . 13 (𝑦 = 𝑐 → ((rec(𝐺, ∅)‘suc 𝑏) “ 𝑦) = ((rec(𝐺, ∅)‘suc 𝑏) “ 𝑐))
109108fveq2d 6846 . . . . . . . . . . . 12 (𝑦 = 𝑐 → (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑦)) = (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑐)))
110 eqid 2736 . . . . . . . . . . . 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)))
111 fvex 6855 . . . . . . . . . . . 12 (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑐)) ∈ V
112109, 110, 111fvmpt 6948 . . . . . . . . . . 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)))
113107, 112syl 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)))
11497, 113eqtrid 2788 . . . . . . . . 9 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → ((𝐺‘(rec(𝐺, ∅)‘suc 𝑏))‘𝑐) = (𝐹‘((rec(𝐺, ∅)‘suc 𝑏) “ 𝑐)))
11558, 86, 1143eqtr4d 2786 . . . . . . . 8 ((𝑏 ∈ ω ∧ 𝑐 ∈ (𝑅1‘suc 𝑏)) → ((𝐺‘((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)))‘𝑐) = ((𝐺‘(rec(𝐺, ∅)‘suc 𝑏))‘𝑐))
116115adantlr 713 . . . . . . 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))
117 fveq2 6842 . . . . . . . . 9 ((rec(𝐺, ∅)‘𝑏) = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) → (𝐺‘(rec(𝐺, ∅)‘𝑏)) = (𝐺‘((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏))))
118117fveq1d 6844 . . . . . . . 8 ((rec(𝐺, ∅)‘𝑏) = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) → ((𝐺‘(rec(𝐺, ∅)‘𝑏))‘𝑐) = ((𝐺‘((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)))‘𝑐))
119118ad2antlr 725 . . . . . . 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))
120 rdgsuc 8370 . . . . . . . . . 10 (suc 𝑏 ∈ On → (rec(𝐺, ∅)‘suc suc 𝑏) = (𝐺‘(rec(𝐺, ∅)‘suc 𝑏)))
12144, 120syl 17 . . . . . . . . 9 (𝑏 ∈ ω → (rec(𝐺, ∅)‘suc suc 𝑏) = (𝐺‘(rec(𝐺, ∅)‘suc 𝑏)))
122121fveq1d 6844 . . . . . . . 8 (𝑏 ∈ ω → ((rec(𝐺, ∅)‘suc suc 𝑏)‘𝑐) = ((𝐺‘(rec(𝐺, ∅)‘suc 𝑏))‘𝑐))
123122ad2antrr 724 . . . . . . 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))
124116, 119, 1233eqtr4rd 2787 . . . . . 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))
125 fvres 6861 . . . . . . 7 (𝑐 ∈ (𝑅1‘suc 𝑏) → (((rec(𝐺, ∅)‘suc suc 𝑏) ↾ (𝑅1‘suc 𝑏))‘𝑐) = ((rec(𝐺, ∅)‘suc suc 𝑏)‘𝑐))
126125adantl 482 . . . . . 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))
127 rdgsuc 8370 . . . . . . . . 9 (𝑏 ∈ On → (rec(𝐺, ∅)‘suc 𝑏) = (𝐺‘(rec(𝐺, ∅)‘𝑏)))
12850, 127syl 17 . . . . . . . 8 (𝑏 ∈ ω → (rec(𝐺, ∅)‘suc 𝑏) = (𝐺‘(rec(𝐺, ∅)‘𝑏)))
129128fveq1d 6844 . . . . . . 7 (𝑏 ∈ ω → ((rec(𝐺, ∅)‘suc 𝑏)‘𝑐) = ((𝐺‘(rec(𝐺, ∅)‘𝑏))‘𝑐))
130129ad2antrr 724 . . . . . 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))
131124, 126, 1303eqtr4rd 2787 . . . . 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))
13238, 49, 131eqfnfvd 6985 . . . 4 ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏) = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏))) → (rec(𝐺, ∅)‘suc 𝑏) = ((rec(𝐺, ∅)‘suc suc 𝑏) ↾ (𝑅1‘suc 𝑏)))
133132ex 413 . . 3 (𝑏 ∈ ω → ((rec(𝐺, ∅)‘𝑏) = ((rec(𝐺, ∅)‘suc 𝑏) ↾ (𝑅1𝑏)) → (rec(𝐺, ∅)‘suc 𝑏) = ((rec(𝐺, ∅)‘suc suc 𝑏) ↾ (𝑅1‘suc 𝑏))))
1346, 12, 18, 24, 30, 133finds 7835 . 2 (𝐴 ∈ ω → (rec(𝐺, ∅)‘𝐴) = ((rec(𝐺, ∅)‘suc 𝐴) ↾ (𝑅1𝐴)))
135 resss 5962 . 2 ((rec(𝐺, ∅)‘suc 𝐴) ↾ (𝑅1𝐴)) ⊆ (rec(𝐺, ∅)‘suc 𝐴)
136134, 135eqsstrdi 3998 1 (𝐴 ∈ ω → (rec(𝐺, ∅)‘𝐴) ⊆ (rec(𝐺, ∅)‘suc 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  cin 3909  wss 3910  c0 4282  𝒫 cpw 4560  {csn 4586   ciun 4954  cmpt 5188  Tr wtr 5222   × cxp 5631  dom cdm 5633  cres 5635  cima 5636  Oncon0 6317  suc csuc 6319   Fn wfn 6491  1-1-ontowf1o 6495  cfv 6496  ωcom 7802  reccrdg 8355  Fincfn 8883  𝑅1cr1 9698  cardccrd 9871
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-r1 9700  df-dju 9837  df-card 9875
This theorem is referenced by:  ackbij2lem4  10178
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