MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfac12lem1 Structured version   Visualization version   GIF version

Theorem dfac12lem1 9218
Description: Lemma for dfac12 9224. (Contributed by Mario Carneiro, 29-May-2015.)
Hypotheses
Ref Expression
dfac12.1 (𝜑𝐴 ∈ On)
dfac12.3 (𝜑𝐹:𝒫 (har‘(𝑅1𝐴))–1-1→On)
dfac12.4 𝐺 = recs((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦))))))
dfac12.5 (𝜑𝐶 ∈ On)
dfac12.h 𝐻 = (OrdIso( E , ran (𝐺 𝐶)) ∘ (𝐺 𝐶))
Assertion
Ref Expression
dfac12lem1 (𝜑 → (𝐺𝐶) = (𝑦 ∈ (𝑅1𝐶) ↦ if(𝐶 = 𝐶, ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻𝑦)))))
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦,𝐶   𝑥,𝐺,𝑦   𝜑,𝑦   𝑥,𝐹,𝑦   𝑦,𝐻
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐻(𝑥)

Proof of Theorem dfac12lem1
StepHypRef Expression
1 dfac12.5 . . 3 (𝜑𝐶 ∈ On)
2 dfac12.4 . . . 4 𝐺 = recs((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦))))))
32tfr2 7698 . . 3 (𝐶 ∈ On → (𝐺𝐶) = ((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦)))))‘(𝐺𝐶)))
41, 3syl 17 . 2 (𝜑 → (𝐺𝐶) = ((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦)))))‘(𝐺𝐶)))
52tfr1 7697 . . . . 5 𝐺 Fn On
6 fnfun 6166 . . . . 5 (𝐺 Fn On → Fun 𝐺)
75, 6ax-mp 5 . . . 4 Fun 𝐺
8 resfunexg 6672 . . . 4 ((Fun 𝐺𝐶 ∈ On) → (𝐺𝐶) ∈ V)
97, 1, 8sylancr 581 . . 3 (𝜑 → (𝐺𝐶) ∈ V)
10 dmeq 5492 . . . . . 6 (𝑥 = (𝐺𝐶) → dom 𝑥 = dom (𝐺𝐶))
1110fveq2d 6379 . . . . 5 (𝑥 = (𝐺𝐶) → (𝑅1‘dom 𝑥) = (𝑅1‘dom (𝐺𝐶)))
1210unieqd 4604 . . . . . . 7 (𝑥 = (𝐺𝐶) → dom 𝑥 = dom (𝐺𝐶))
1310, 12eqeq12d 2780 . . . . . 6 (𝑥 = (𝐺𝐶) → (dom 𝑥 = dom 𝑥 ↔ dom (𝐺𝐶) = dom (𝐺𝐶)))
14 rneq 5519 . . . . . . . . . . . . 13 (𝑥 = (𝐺𝐶) → ran 𝑥 = ran (𝐺𝐶))
15 df-ima 5290 . . . . . . . . . . . . 13 (𝐺𝐶) = ran (𝐺𝐶)
1614, 15syl6eqr 2817 . . . . . . . . . . . 12 (𝑥 = (𝐺𝐶) → ran 𝑥 = (𝐺𝐶))
1716unieqd 4604 . . . . . . . . . . 11 (𝑥 = (𝐺𝐶) → ran 𝑥 = (𝐺𝐶))
1817rneqd 5521 . . . . . . . . . 10 (𝑥 = (𝐺𝐶) → ran ran 𝑥 = ran (𝐺𝐶))
1918unieqd 4604 . . . . . . . . 9 (𝑥 = (𝐺𝐶) → ran ran 𝑥 = ran (𝐺𝐶))
20 suceq 5973 . . . . . . . . 9 ( ran ran 𝑥 = ran (𝐺𝐶) → suc ran ran 𝑥 = suc ran (𝐺𝐶))
2119, 20syl 17 . . . . . . . 8 (𝑥 = (𝐺𝐶) → suc ran ran 𝑥 = suc ran (𝐺𝐶))
2221oveq1d 6857 . . . . . . 7 (𝑥 = (𝐺𝐶) → (suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) = (suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)))
23 fveq1 6374 . . . . . . . 8 (𝑥 = (𝐺𝐶) → (𝑥‘suc (rank‘𝑦)) = ((𝐺𝐶)‘suc (rank‘𝑦)))
2423fveq1d 6377 . . . . . . 7 (𝑥 = (𝐺𝐶) → ((𝑥‘suc (rank‘𝑦))‘𝑦) = (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦))
2522, 24oveq12d 6860 . . . . . 6 (𝑥 = (𝐺𝐶) → ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)) = ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)))
26 id 22 . . . . . . . . . . . . 13 (𝑥 = (𝐺𝐶) → 𝑥 = (𝐺𝐶))
2726, 12fveq12d 6382 . . . . . . . . . . . 12 (𝑥 = (𝐺𝐶) → (𝑥 dom 𝑥) = ((𝐺𝐶)‘ dom (𝐺𝐶)))
2827rneqd 5521 . . . . . . . . . . 11 (𝑥 = (𝐺𝐶) → ran (𝑥 dom 𝑥) = ran ((𝐺𝐶)‘ dom (𝐺𝐶)))
29 oieq2 8625 . . . . . . . . . . 11 (ran (𝑥 dom 𝑥) = ran ((𝐺𝐶)‘ dom (𝐺𝐶)) → OrdIso( E , ran (𝑥 dom 𝑥)) = OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))))
3028, 29syl 17 . . . . . . . . . 10 (𝑥 = (𝐺𝐶) → OrdIso( E , ran (𝑥 dom 𝑥)) = OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))))
3130cnveqd 5466 . . . . . . . . 9 (𝑥 = (𝐺𝐶) → OrdIso( E , ran (𝑥 dom 𝑥)) = OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))))
3231, 27coeq12d 5455 . . . . . . . 8 (𝑥 = (𝐺𝐶) → (OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) = (OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))))
3332imaeq1d 5647 . . . . . . 7 (𝑥 = (𝐺𝐶) → ((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦) = ((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦))
3433fveq2d 6379 . . . . . 6 (𝑥 = (𝐺𝐶) → (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦)) = (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦)))
3513, 25, 34ifbieq12d 4270 . . . . 5 (𝑥 = (𝐺𝐶) → if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦))) = if(dom (𝐺𝐶) = dom (𝐺𝐶), ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦))))
3611, 35mpteq12dv 4892 . . . 4 (𝑥 = (𝐺𝐶) → (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦)))) = (𝑦 ∈ (𝑅1‘dom (𝐺𝐶)) ↦ if(dom (𝐺𝐶) = dom (𝐺𝐶), ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦)))))
37 eqid 2765 . . . 4 (𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦))))) = (𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦)))))
38 fvex 6388 . . . . 5 (𝑅1‘dom (𝐺𝐶)) ∈ V
3938mptex 6679 . . . 4 (𝑦 ∈ (𝑅1‘dom (𝐺𝐶)) ↦ if(dom (𝐺𝐶) = dom (𝐺𝐶), ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦)))) ∈ V
4036, 37, 39fvmpt 6471 . . 3 ((𝐺𝐶) ∈ V → ((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦)))))‘(𝐺𝐶)) = (𝑦 ∈ (𝑅1‘dom (𝐺𝐶)) ↦ if(dom (𝐺𝐶) = dom (𝐺𝐶), ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦)))))
419, 40syl 17 . 2 (𝜑 → ((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦)))))‘(𝐺𝐶)) = (𝑦 ∈ (𝑅1‘dom (𝐺𝐶)) ↦ if(dom (𝐺𝐶) = dom (𝐺𝐶), ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦)))))
42 onss 7188 . . . . . . . 8 (𝐶 ∈ On → 𝐶 ⊆ On)
431, 42syl 17 . . . . . . 7 (𝜑𝐶 ⊆ On)
44 fnssres 6182 . . . . . . 7 ((𝐺 Fn On ∧ 𝐶 ⊆ On) → (𝐺𝐶) Fn 𝐶)
455, 43, 44sylancr 581 . . . . . 6 (𝜑 → (𝐺𝐶) Fn 𝐶)
46 fndm 6168 . . . . . 6 ((𝐺𝐶) Fn 𝐶 → dom (𝐺𝐶) = 𝐶)
4745, 46syl 17 . . . . 5 (𝜑 → dom (𝐺𝐶) = 𝐶)
4847fveq2d 6379 . . . 4 (𝜑 → (𝑅1‘dom (𝐺𝐶)) = (𝑅1𝐶))
4948mpteq1d 4897 . . 3 (𝜑 → (𝑦 ∈ (𝑅1‘dom (𝐺𝐶)) ↦ if(dom (𝐺𝐶) = dom (𝐺𝐶), ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦)))) = (𝑦 ∈ (𝑅1𝐶) ↦ if(dom (𝐺𝐶) = dom (𝐺𝐶), ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦)))))
5047adantr 472 . . . . . . 7 ((𝜑𝑦 ∈ (𝑅1𝐶)) → dom (𝐺𝐶) = 𝐶)
5150unieqd 4604 . . . . . . 7 ((𝜑𝑦 ∈ (𝑅1𝐶)) → dom (𝐺𝐶) = 𝐶)
5250, 51eqeq12d 2780 . . . . . 6 ((𝜑𝑦 ∈ (𝑅1𝐶)) → (dom (𝐺𝐶) = dom (𝐺𝐶) ↔ 𝐶 = 𝐶))
5352ifbid 4265 . . . . 5 ((𝜑𝑦 ∈ (𝑅1𝐶)) → if(dom (𝐺𝐶) = dom (𝐺𝐶), ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦))) = if(𝐶 = 𝐶, ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦))))
54 rankr1ai 8876 . . . . . . . . . . . 12 (𝑦 ∈ (𝑅1𝐶) → (rank‘𝑦) ∈ 𝐶)
5554ad2antlr 718 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → (rank‘𝑦) ∈ 𝐶)
56 simpr 477 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → 𝐶 = 𝐶)
5755, 56eleqtrd 2846 . . . . . . . . . 10 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → (rank‘𝑦) ∈ 𝐶)
58 eloni 5918 . . . . . . . . . . . 12 (𝐶 ∈ On → Ord 𝐶)
59 ordsucuniel 7222 . . . . . . . . . . . 12 (Ord 𝐶 → ((rank‘𝑦) ∈ 𝐶 ↔ suc (rank‘𝑦) ∈ 𝐶))
601, 58, 593syl 18 . . . . . . . . . . 11 (𝜑 → ((rank‘𝑦) ∈ 𝐶 ↔ suc (rank‘𝑦) ∈ 𝐶))
6160ad2antrr 717 . . . . . . . . . 10 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → ((rank‘𝑦) ∈ 𝐶 ↔ suc (rank‘𝑦) ∈ 𝐶))
6257, 61mpbid 223 . . . . . . . . 9 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → suc (rank‘𝑦) ∈ 𝐶)
63 fvres 6394 . . . . . . . . 9 (suc (rank‘𝑦) ∈ 𝐶 → ((𝐺𝐶)‘suc (rank‘𝑦)) = (𝐺‘suc (rank‘𝑦)))
6462, 63syl 17 . . . . . . . 8 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → ((𝐺𝐶)‘suc (rank‘𝑦)) = (𝐺‘suc (rank‘𝑦)))
6564fveq1d 6377 . . . . . . 7 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑦))‘𝑦))
6665oveq2d 6858 . . . . . 6 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)) = ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)))
6766ifeq1da 4273 . . . . 5 ((𝜑𝑦 ∈ (𝑅1𝐶)) → if(𝐶 = 𝐶, ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦))) = if(𝐶 = 𝐶, ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦))))
6851adantr 472 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → dom (𝐺𝐶) = 𝐶)
6968fveq2d 6379 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → ((𝐺𝐶)‘ dom (𝐺𝐶)) = ((𝐺𝐶)‘ 𝐶))
701ad2antrr 717 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → 𝐶 ∈ On)
71 uniexg 7153 . . . . . . . . . . . . . . . . 17 (𝐶 ∈ On → 𝐶 ∈ V)
72 sucidg 5986 . . . . . . . . . . . . . . . . 17 ( 𝐶 ∈ V → 𝐶 ∈ suc 𝐶)
7370, 71, 723syl 18 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → 𝐶 ∈ suc 𝐶)
741adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (𝑅1𝐶)) → 𝐶 ∈ On)
75 orduniorsuc 7228 . . . . . . . . . . . . . . . . . 18 (Ord 𝐶 → (𝐶 = 𝐶𝐶 = suc 𝐶))
7674, 58, 753syl 18 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (𝑅1𝐶)) → (𝐶 = 𝐶𝐶 = suc 𝐶))
7776orcanai 1025 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → 𝐶 = suc 𝐶)
7873, 77eleqtrrd 2847 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → 𝐶𝐶)
79 fvres 6394 . . . . . . . . . . . . . . 15 ( 𝐶𝐶 → ((𝐺𝐶)‘ 𝐶) = (𝐺 𝐶))
8078, 79syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → ((𝐺𝐶)‘ 𝐶) = (𝐺 𝐶))
8169, 80eqtrd 2799 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → ((𝐺𝐶)‘ dom (𝐺𝐶)) = (𝐺 𝐶))
8281rneqd 5521 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → ran ((𝐺𝐶)‘ dom (𝐺𝐶)) = ran (𝐺 𝐶))
83 oieq2 8625 . . . . . . . . . . . 12 (ran ((𝐺𝐶)‘ dom (𝐺𝐶)) = ran (𝐺 𝐶) → OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) = OrdIso( E , ran (𝐺 𝐶)))
8482, 83syl 17 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) = OrdIso( E , ran (𝐺 𝐶)))
8584cnveqd 5466 . . . . . . . . . 10 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) = OrdIso( E , ran (𝐺 𝐶)))
8685, 81coeq12d 5455 . . . . . . . . 9 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → (OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) = (OrdIso( E , ran (𝐺 𝐶)) ∘ (𝐺 𝐶)))
87 dfac12.h . . . . . . . . 9 𝐻 = (OrdIso( E , ran (𝐺 𝐶)) ∘ (𝐺 𝐶))
8886, 87syl6eqr 2817 . . . . . . . 8 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → (OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) = 𝐻)
8988imaeq1d 5647 . . . . . . 7 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → ((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦) = (𝐻𝑦))
9089fveq2d 6379 . . . . . 6 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦)) = (𝐹‘(𝐻𝑦)))
9190ifeq2da 4274 . . . . 5 ((𝜑𝑦 ∈ (𝑅1𝐶)) → if(𝐶 = 𝐶, ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦))) = if(𝐶 = 𝐶, ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻𝑦))))
9253, 67, 913eqtrd 2803 . . . 4 ((𝜑𝑦 ∈ (𝑅1𝐶)) → if(dom (𝐺𝐶) = dom (𝐺𝐶), ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦))) = if(𝐶 = 𝐶, ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻𝑦))))
9392mpteq2dva 4903 . . 3 (𝜑 → (𝑦 ∈ (𝑅1𝐶) ↦ if(dom (𝐺𝐶) = dom (𝐺𝐶), ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦)))) = (𝑦 ∈ (𝑅1𝐶) ↦ if(𝐶 = 𝐶, ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻𝑦)))))
9449, 93eqtrd 2799 . 2 (𝜑 → (𝑦 ∈ (𝑅1‘dom (𝐺𝐶)) ↦ if(dom (𝐺𝐶) = dom (𝐺𝐶), ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦)))) = (𝑦 ∈ (𝑅1𝐶) ↦ if(𝐶 = 𝐶, ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻𝑦)))))
954, 41, 943eqtrd 2803 1 (𝜑 → (𝐺𝐶) = (𝑦 ∈ (𝑅1𝐶) ↦ if(𝐶 = 𝐶, ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873   = wceq 1652  wcel 2155  Vcvv 3350  wss 3732  ifcif 4243  𝒫 cpw 4315   cuni 4594  cmpt 4888   E cep 5189  ccnv 5276  dom cdm 5277  ran crn 5278  cres 5279  cima 5280  ccom 5281  Ord word 5907  Oncon0 5908  suc csuc 5910  Fun wfun 6062   Fn wfn 6063  1-1wf1 6065  cfv 6068  (class class class)co 6842  recscrecs 7671   +𝑜 coa 7761   ·𝑜 comu 7762  OrdIsocoi 8621  harchar 8668  𝑅1cr1 8840  rankcrnk 8841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-oi 8622  df-r1 8842  df-rank 8843
This theorem is referenced by:  dfac12lem2  9219
  Copyright terms: Public domain W3C validator