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Theorem ackbij2lem2 9009
Description: Lemma for ackbij2 9012. (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
ackbij2lem2 (𝐴 ∈ ω → (rec(𝐺, ∅)‘𝐴):(𝑅1𝐴)–1-1-onto→(card‘(𝑅1𝐴)))
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝐴,𝑦

Proof of Theorem ackbij2lem2
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6150 . . 3 (𝑎 = ∅ → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘∅))
2 fveq2 6150 . . 3 (𝑎 = ∅ → (𝑅1𝑎) = (𝑅1‘∅))
32fveq2d 6154 . . 3 (𝑎 = ∅ → (card‘(𝑅1𝑎)) = (card‘(𝑅1‘∅)))
41, 2, 3f1oeq123d 6092 . 2 (𝑎 = ∅ → ((rec(𝐺, ∅)‘𝑎):(𝑅1𝑎)–1-1-onto→(card‘(𝑅1𝑎)) ↔ (rec(𝐺, ∅)‘∅):(𝑅1‘∅)–1-1-onto→(card‘(𝑅1‘∅))))
5 fveq2 6150 . . 3 (𝑎 = 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝑏))
6 fveq2 6150 . . 3 (𝑎 = 𝑏 → (𝑅1𝑎) = (𝑅1𝑏))
76fveq2d 6154 . . 3 (𝑎 = 𝑏 → (card‘(𝑅1𝑎)) = (card‘(𝑅1𝑏)))
85, 6, 7f1oeq123d 6092 . 2 (𝑎 = 𝑏 → ((rec(𝐺, ∅)‘𝑎):(𝑅1𝑎)–1-1-onto→(card‘(𝑅1𝑎)) ↔ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))))
9 fveq2 6150 . . 3 (𝑎 = suc 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘suc 𝑏))
10 fveq2 6150 . . 3 (𝑎 = suc 𝑏 → (𝑅1𝑎) = (𝑅1‘suc 𝑏))
1110fveq2d 6154 . . 3 (𝑎 = suc 𝑏 → (card‘(𝑅1𝑎)) = (card‘(𝑅1‘suc 𝑏)))
129, 10, 11f1oeq123d 6092 . 2 (𝑎 = suc 𝑏 → ((rec(𝐺, ∅)‘𝑎):(𝑅1𝑎)–1-1-onto→(card‘(𝑅1𝑎)) ↔ (rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏))))
13 fveq2 6150 . . 3 (𝑎 = 𝐴 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝐴))
14 fveq2 6150 . . 3 (𝑎 = 𝐴 → (𝑅1𝑎) = (𝑅1𝐴))
1514fveq2d 6154 . . 3 (𝑎 = 𝐴 → (card‘(𝑅1𝑎)) = (card‘(𝑅1𝐴)))
1613, 14, 15f1oeq123d 6092 . 2 (𝑎 = 𝐴 → ((rec(𝐺, ∅)‘𝑎):(𝑅1𝑎)–1-1-onto→(card‘(𝑅1𝑎)) ↔ (rec(𝐺, ∅)‘𝐴):(𝑅1𝐴)–1-1-onto→(card‘(𝑅1𝐴))))
17 f1o0 6132 . . 3 ∅:∅–1-1-onto→∅
18 0ex 4752 . . . . . 6 ∅ ∈ V
1918rdg0 7465 . . . . 5 (rec(𝐺, ∅)‘∅) = ∅
20 f1oeq1 6086 . . . . 5 ((rec(𝐺, ∅)‘∅) = ∅ → ((rec(𝐺, ∅)‘∅):(𝑅1‘∅)–1-1-onto→(card‘(𝑅1‘∅)) ↔ ∅:(𝑅1‘∅)–1-1-onto→(card‘(𝑅1‘∅))))
2119, 20ax-mp 5 . . . 4 ((rec(𝐺, ∅)‘∅):(𝑅1‘∅)–1-1-onto→(card‘(𝑅1‘∅)) ↔ ∅:(𝑅1‘∅)–1-1-onto→(card‘(𝑅1‘∅)))
22 r10 8578 . . . . 5 (𝑅1‘∅) = ∅
2322fveq2i 6153 . . . . . 6 (card‘(𝑅1‘∅)) = (card‘∅)
24 card0 8731 . . . . . 6 (card‘∅) = ∅
2523, 24eqtri 2643 . . . . 5 (card‘(𝑅1‘∅)) = ∅
26 f1oeq23 6089 . . . . 5 (((𝑅1‘∅) = ∅ ∧ (card‘(𝑅1‘∅)) = ∅) → (∅:(𝑅1‘∅)–1-1-onto→(card‘(𝑅1‘∅)) ↔ ∅:∅–1-1-onto→∅))
2722, 25, 26mp2an 707 . . . 4 (∅:(𝑅1‘∅)–1-1-onto→(card‘(𝑅1‘∅)) ↔ ∅:∅–1-1-onto→∅)
2821, 27bitri 264 . . 3 ((rec(𝐺, ∅)‘∅):(𝑅1‘∅)–1-1-onto→(card‘(𝑅1‘∅)) ↔ ∅:∅–1-1-onto→∅)
2917, 28mpbir 221 . 2 (rec(𝐺, ∅)‘∅):(𝑅1‘∅)–1-1-onto→(card‘(𝑅1‘∅))
30 ackbij.f . . . . . . . . . 10 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
3130ackbij1lem17 9005 . . . . . . . . 9 𝐹:(𝒫 ω ∩ Fin)–1-1→ω
3231a1i 11 . . . . . . . 8 (𝑏 ∈ ω → 𝐹:(𝒫 ω ∩ Fin)–1-1→ω)
33 r1fin 8583 . . . . . . . . . 10 (𝑏 ∈ ω → (𝑅1𝑏) ∈ Fin)
34 ficardom 8734 . . . . . . . . . 10 ((𝑅1𝑏) ∈ Fin → (card‘(𝑅1𝑏)) ∈ ω)
3533, 34syl 17 . . . . . . . . 9 (𝑏 ∈ ω → (card‘(𝑅1𝑏)) ∈ ω)
36 ackbij2lem1 8988 . . . . . . . . 9 ((card‘(𝑅1𝑏)) ∈ ω → 𝒫 (card‘(𝑅1𝑏)) ⊆ (𝒫 ω ∩ Fin))
3735, 36syl 17 . . . . . . . 8 (𝑏 ∈ ω → 𝒫 (card‘(𝑅1𝑏)) ⊆ (𝒫 ω ∩ Fin))
38 f1ores 6110 . . . . . . . 8 ((𝐹:(𝒫 ω ∩ Fin)–1-1→ω ∧ 𝒫 (card‘(𝑅1𝑏)) ⊆ (𝒫 ω ∩ Fin)) → (𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))):𝒫 (card‘(𝑅1𝑏))–1-1-onto→(𝐹 “ 𝒫 (card‘(𝑅1𝑏))))
3932, 37, 38syl2anc 692 . . . . . . 7 (𝑏 ∈ ω → (𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))):𝒫 (card‘(𝑅1𝑏))–1-1-onto→(𝐹 “ 𝒫 (card‘(𝑅1𝑏))))
4030ackbij1b 9008 . . . . . . . . . 10 ((card‘(𝑅1𝑏)) ∈ ω → (𝐹 “ 𝒫 (card‘(𝑅1𝑏))) = (card‘𝒫 (card‘(𝑅1𝑏))))
4135, 40syl 17 . . . . . . . . 9 (𝑏 ∈ ω → (𝐹 “ 𝒫 (card‘(𝑅1𝑏))) = (card‘𝒫 (card‘(𝑅1𝑏))))
42 ficardid 8735 . . . . . . . . . 10 ((𝑅1𝑏) ∈ Fin → (card‘(𝑅1𝑏)) ≈ (𝑅1𝑏))
43 pwen 8080 . . . . . . . . . 10 ((card‘(𝑅1𝑏)) ≈ (𝑅1𝑏) → 𝒫 (card‘(𝑅1𝑏)) ≈ 𝒫 (𝑅1𝑏))
44 carden2b 8740 . . . . . . . . . 10 (𝒫 (card‘(𝑅1𝑏)) ≈ 𝒫 (𝑅1𝑏) → (card‘𝒫 (card‘(𝑅1𝑏))) = (card‘𝒫 (𝑅1𝑏)))
4533, 42, 43, 444syl 19 . . . . . . . . 9 (𝑏 ∈ ω → (card‘𝒫 (card‘(𝑅1𝑏))) = (card‘𝒫 (𝑅1𝑏)))
4641, 45eqtrd 2655 . . . . . . . 8 (𝑏 ∈ ω → (𝐹 “ 𝒫 (card‘(𝑅1𝑏))) = (card‘𝒫 (𝑅1𝑏)))
4746f1oeq3d 6093 . . . . . . 7 (𝑏 ∈ ω → ((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))):𝒫 (card‘(𝑅1𝑏))–1-1-onto→(𝐹 “ 𝒫 (card‘(𝑅1𝑏))) ↔ (𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))):𝒫 (card‘(𝑅1𝑏))–1-1-onto→(card‘𝒫 (𝑅1𝑏))))
4839, 47mpbid 222 . . . . . 6 (𝑏 ∈ ω → (𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))):𝒫 (card‘(𝑅1𝑏))–1-1-onto→(card‘𝒫 (𝑅1𝑏)))
4948adantr 481 . . . . 5 ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) → (𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))):𝒫 (card‘(𝑅1𝑏))–1-1-onto→(card‘𝒫 (𝑅1𝑏)))
50 f1opw 6845 . . . . . 6 ((rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏)) → (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)):𝒫 (𝑅1𝑏)–1-1-onto→𝒫 (card‘(𝑅1𝑏)))
5150adantl 482 . . . . 5 ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) → (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)):𝒫 (𝑅1𝑏)–1-1-onto→𝒫 (card‘(𝑅1𝑏)))
52 f1oco 6118 . . . . 5 (((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))):𝒫 (card‘(𝑅1𝑏))–1-1-onto→(card‘𝒫 (𝑅1𝑏)) ∧ (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)):𝒫 (𝑅1𝑏)–1-1-onto→𝒫 (card‘(𝑅1𝑏))) → ((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):𝒫 (𝑅1𝑏)–1-1-onto→(card‘𝒫 (𝑅1𝑏)))
5349, 51, 52syl2anc 692 . . . 4 ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) → ((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):𝒫 (𝑅1𝑏)–1-1-onto→(card‘𝒫 (𝑅1𝑏)))
54 frsuc 7480 . . . . . . . . 9 (𝑏 ∈ ω → ((rec(𝐺, ∅) ↾ ω)‘suc 𝑏) = (𝐺‘((rec(𝐺, ∅) ↾ ω)‘𝑏)))
55 peano2 7036 . . . . . . . . . 10 (𝑏 ∈ ω → suc 𝑏 ∈ ω)
5655fvresd 6167 . . . . . . . . 9 (𝑏 ∈ ω → ((rec(𝐺, ∅) ↾ ω)‘suc 𝑏) = (rec(𝐺, ∅)‘suc 𝑏))
57 fvres 6166 . . . . . . . . . . 11 (𝑏 ∈ ω → ((rec(𝐺, ∅) ↾ ω)‘𝑏) = (rec(𝐺, ∅)‘𝑏))
5857fveq2d 6154 . . . . . . . . . 10 (𝑏 ∈ ω → (𝐺‘((rec(𝐺, ∅) ↾ ω)‘𝑏)) = (𝐺‘(rec(𝐺, ∅)‘𝑏)))
59 fvex 6160 . . . . . . . . . . 11 (rec(𝐺, ∅)‘𝑏) ∈ V
60 dmeq 5286 . . . . . . . . . . . . . 14 (𝑥 = (rec(𝐺, ∅)‘𝑏) → dom 𝑥 = dom (rec(𝐺, ∅)‘𝑏))
6160pweqd 4137 . . . . . . . . . . . . 13 (𝑥 = (rec(𝐺, ∅)‘𝑏) → 𝒫 dom 𝑥 = 𝒫 dom (rec(𝐺, ∅)‘𝑏))
62 imaeq1 5422 . . . . . . . . . . . . . 14 (𝑥 = (rec(𝐺, ∅)‘𝑏) → (𝑥𝑦) = ((rec(𝐺, ∅)‘𝑏) “ 𝑦))
6362fveq2d 6154 . . . . . . . . . . . . 13 (𝑥 = (rec(𝐺, ∅)‘𝑏) → (𝐹‘(𝑥𝑦)) = (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)))
6461, 63mpteq12dv 4695 . . . . . . . . . . . 12 (𝑥 = (rec(𝐺, ∅)‘𝑏) → (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥𝑦))) = (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))))
65 ackbij.g . . . . . . . . . . . 12 𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥𝑦))))
6659dmex 7049 . . . . . . . . . . . . . 14 dom (rec(𝐺, ∅)‘𝑏) ∈ V
6766pwex 4810 . . . . . . . . . . . . 13 𝒫 dom (rec(𝐺, ∅)‘𝑏) ∈ V
6867mptex 6443 . . . . . . . . . . . 12 (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))) ∈ V
6964, 65, 68fvmpt 6241 . . . . . . . . . . 11 ((rec(𝐺, ∅)‘𝑏) ∈ V → (𝐺‘(rec(𝐺, ∅)‘𝑏)) = (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))))
7059, 69ax-mp 5 . . . . . . . . . 10 (𝐺‘(rec(𝐺, ∅)‘𝑏)) = (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)))
7158, 70syl6eq 2671 . . . . . . . . 9 (𝑏 ∈ ω → (𝐺‘((rec(𝐺, ∅) ↾ ω)‘𝑏)) = (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))))
7254, 56, 713eqtr3d 2663 . . . . . . . 8 (𝑏 ∈ ω → (rec(𝐺, ∅)‘suc 𝑏) = (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))))
7372adantr 481 . . . . . . 7 ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) → (rec(𝐺, ∅)‘suc 𝑏) = (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))))
74 f1odm 6100 . . . . . . . . . . 11 ((rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏)) → dom (rec(𝐺, ∅)‘𝑏) = (𝑅1𝑏))
7574adantl 482 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) → dom (rec(𝐺, ∅)‘𝑏) = (𝑅1𝑏))
7675pweqd 4137 . . . . . . . . 9 ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) → 𝒫 dom (rec(𝐺, ∅)‘𝑏) = 𝒫 (𝑅1𝑏))
7776mpteq1d 4700 . . . . . . . 8 ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) → (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))) = (𝑦 ∈ 𝒫 (𝑅1𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))))
78 fvex 6160 . . . . . . . . . . 11 (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)) ∈ V
79 eqid 2621 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 (𝑅1𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))) = (𝑦 ∈ 𝒫 (𝑅1𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)))
8078, 79fnmpti 5981 . . . . . . . . . 10 (𝑦 ∈ 𝒫 (𝑅1𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))) Fn 𝒫 (𝑅1𝑏)
8180a1i 11 . . . . . . . . 9 ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) → (𝑦 ∈ 𝒫 (𝑅1𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))) Fn 𝒫 (𝑅1𝑏))
82 f1ofn 6097 . . . . . . . . . 10 (((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):𝒫 (𝑅1𝑏)–1-1-onto→(card‘𝒫 (𝑅1𝑏)) → ((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))) Fn 𝒫 (𝑅1𝑏))
8353, 82syl 17 . . . . . . . . 9 ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) → ((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))) Fn 𝒫 (𝑅1𝑏))
84 f1of 6096 . . . . . . . . . . . . . 14 ((𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)):𝒫 (𝑅1𝑏)–1-1-onto→𝒫 (card‘(𝑅1𝑏)) → (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)):𝒫 (𝑅1𝑏)⟶𝒫 (card‘(𝑅1𝑏)))
8551, 84syl 17 . . . . . . . . . . . . 13 ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) → (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)):𝒫 (𝑅1𝑏)⟶𝒫 (card‘(𝑅1𝑏)))
8685ffvelrnda 6317 . . . . . . . . . . . 12 (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) ∧ 𝑐 ∈ 𝒫 (𝑅1𝑏)) → ((𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))‘𝑐) ∈ 𝒫 (card‘(𝑅1𝑏)))
8786fvresd 6167 . . . . . . . . . . 11 (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) ∧ 𝑐 ∈ 𝒫 (𝑅1𝑏)) → ((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏)))‘((𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))‘𝑐)) = (𝐹‘((𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))‘𝑐)))
88 imaeq2 5423 . . . . . . . . . . . . . 14 (𝑎 = 𝑐 → ((rec(𝐺, ∅)‘𝑏) “ 𝑎) = ((rec(𝐺, ∅)‘𝑏) “ 𝑐))
89 eqid 2621 . . . . . . . . . . . . . 14 (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)) = (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))
9059imaex 7054 . . . . . . . . . . . . . 14 ((rec(𝐺, ∅)‘𝑏) “ 𝑐) ∈ V
9188, 89, 90fvmpt 6241 . . . . . . . . . . . . 13 (𝑐 ∈ 𝒫 (𝑅1𝑏) → ((𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))‘𝑐) = ((rec(𝐺, ∅)‘𝑏) “ 𝑐))
9291adantl 482 . . . . . . . . . . . 12 (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) ∧ 𝑐 ∈ 𝒫 (𝑅1𝑏)) → ((𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))‘𝑐) = ((rec(𝐺, ∅)‘𝑏) “ 𝑐))
9392fveq2d 6154 . . . . . . . . . . 11 (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) ∧ 𝑐 ∈ 𝒫 (𝑅1𝑏)) → (𝐹‘((𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))‘𝑐)) = (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑐)))
9487, 93eqtrd 2655 . . . . . . . . . 10 (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) ∧ 𝑐 ∈ 𝒫 (𝑅1𝑏)) → ((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏)))‘((𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))‘𝑐)) = (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑐)))
95 fvco3 6234 . . . . . . . . . . 11 (((𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)):𝒫 (𝑅1𝑏)⟶𝒫 (card‘(𝑅1𝑏)) ∧ 𝑐 ∈ 𝒫 (𝑅1𝑏)) → (((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)))‘𝑐) = ((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏)))‘((𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))‘𝑐)))
9685, 95sylan 488 . . . . . . . . . 10 (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) ∧ 𝑐 ∈ 𝒫 (𝑅1𝑏)) → (((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)))‘𝑐) = ((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏)))‘((𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))‘𝑐)))
97 imaeq2 5423 . . . . . . . . . . . . 13 (𝑦 = 𝑐 → ((rec(𝐺, ∅)‘𝑏) “ 𝑦) = ((rec(𝐺, ∅)‘𝑏) “ 𝑐))
9897fveq2d 6154 . . . . . . . . . . . 12 (𝑦 = 𝑐 → (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)) = (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑐)))
99 fvex 6160 . . . . . . . . . . . 12 (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑐)) ∈ V
10098, 79, 99fvmpt 6241 . . . . . . . . . . 11 (𝑐 ∈ 𝒫 (𝑅1𝑏) → ((𝑦 ∈ 𝒫 (𝑅1𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)))‘𝑐) = (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑐)))
101100adantl 482 . . . . . . . . . 10 (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) ∧ 𝑐 ∈ 𝒫 (𝑅1𝑏)) → ((𝑦 ∈ 𝒫 (𝑅1𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)))‘𝑐) = (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑐)))
10294, 96, 1013eqtr4rd 2666 . . . . . . . . 9 (((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) ∧ 𝑐 ∈ 𝒫 (𝑅1𝑏)) → ((𝑦 ∈ 𝒫 (𝑅1𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦)))‘𝑐) = (((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎)))‘𝑐))
10381, 83, 102eqfnfvd 6272 . . . . . . . 8 ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) → (𝑦 ∈ 𝒫 (𝑅1𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))) = ((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))))
10477, 103eqtrd 2655 . . . . . . 7 ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) → (𝑦 ∈ 𝒫 dom (rec(𝐺, ∅)‘𝑏) ↦ (𝐹‘((rec(𝐺, ∅)‘𝑏) “ 𝑦))) = ((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))))
10573, 104eqtrd 2655 . . . . . 6 ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) → (rec(𝐺, ∅)‘suc 𝑏) = ((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))))
106 f1oeq1 6086 . . . . . 6 ((rec(𝐺, ∅)‘suc 𝑏) = ((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))) → ((rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)) ↔ ((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏))))
107105, 106syl 17 . . . . 5 ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) → ((rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)) ↔ ((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏))))
108 nnon 7021 . . . . . . . 8 (𝑏 ∈ ω → 𝑏 ∈ On)
109 r1suc 8580 . . . . . . . 8 (𝑏 ∈ On → (𝑅1‘suc 𝑏) = 𝒫 (𝑅1𝑏))
110108, 109syl 17 . . . . . . 7 (𝑏 ∈ ω → (𝑅1‘suc 𝑏) = 𝒫 (𝑅1𝑏))
111110fveq2d 6154 . . . . . . 7 (𝑏 ∈ ω → (card‘(𝑅1‘suc 𝑏)) = (card‘𝒫 (𝑅1𝑏)))
112 f1oeq23 6089 . . . . . . 7 (((𝑅1‘suc 𝑏) = 𝒫 (𝑅1𝑏) ∧ (card‘(𝑅1‘suc 𝑏)) = (card‘𝒫 (𝑅1𝑏))) → (((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)) ↔ ((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):𝒫 (𝑅1𝑏)–1-1-onto→(card‘𝒫 (𝑅1𝑏))))
113110, 111, 112syl2anc 692 . . . . . 6 (𝑏 ∈ ω → (((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)) ↔ ((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):𝒫 (𝑅1𝑏)–1-1-onto→(card‘𝒫 (𝑅1𝑏))))
114113adantr 481 . . . . 5 ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) → (((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)) ↔ ((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):𝒫 (𝑅1𝑏)–1-1-onto→(card‘𝒫 (𝑅1𝑏))))
115107, 114bitrd 268 . . . 4 ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) → ((rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)) ↔ ((𝐹 ↾ 𝒫 (card‘(𝑅1𝑏))) ∘ (𝑎 ∈ 𝒫 (𝑅1𝑏) ↦ ((rec(𝐺, ∅)‘𝑏) “ 𝑎))):𝒫 (𝑅1𝑏)–1-1-onto→(card‘𝒫 (𝑅1𝑏))))
11653, 115mpbird 247 . . 3 ((𝑏 ∈ ω ∧ (rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏))) → (rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)))
117116ex 450 . 2 (𝑏 ∈ ω → ((rec(𝐺, ∅)‘𝑏):(𝑅1𝑏)–1-1-onto→(card‘(𝑅1𝑏)) → (rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏))))
1184, 8, 12, 16, 29, 117finds 7042 1 (𝐴 ∈ ω → (rec(𝐺, ∅)‘𝐴):(𝑅1𝐴)–1-1-onto→(card‘(𝑅1𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cin 3555  wss 3556  c0 3893  𝒫 cpw 4132  {csn 4150   ciun 4487   class class class wbr 4615  cmpt 4675   × cxp 5074  dom cdm 5076  cres 5078  cima 5079  ccom 5080  Oncon0 5684  suc csuc 5686   Fn wfn 5844  wf 5845  1-1wf1 5846  1-1-ontowf1o 5848  cfv 5849  ωcom 7015  reccrdg 7453  cen 7899  Fincfn 7902  𝑅1cr1 8572  cardccrd 8708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-2o 7509  df-oadd 7512  df-er 7690  df-map 7807  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-r1 8574  df-card 8712  df-cda 8937
This theorem is referenced by:  ackbij2lem3  9010  ackbij2  9012
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