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Theorem dfac12lem1 10079
Description: Lemma for dfac12 10085. (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 8344 . . 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 8343 . . . . 5 𝐺 Fn On
6 fnfun 6602 . . . . 5 (𝐺 Fn On → Fun 𝐺)
75, 6ax-mp 5 . . . 4 Fun 𝐺
8 resfunexg 7165 . . . 4 ((Fun 𝐺𝐶 ∈ On) → (𝐺𝐶) ∈ V)
97, 1, 8sylancr 587 . . 3 (𝜑 → (𝐺𝐶) ∈ V)
10 dmeq 5859 . . . . . 6 (𝑥 = (𝐺𝐶) → dom 𝑥 = dom (𝐺𝐶))
1110fveq2d 6846 . . . . 5 (𝑥 = (𝐺𝐶) → (𝑅1‘dom 𝑥) = (𝑅1‘dom (𝐺𝐶)))
1210unieqd 4879 . . . . . . 7 (𝑥 = (𝐺𝐶) → dom 𝑥 = dom (𝐺𝐶))
1310, 12eqeq12d 2752 . . . . . 6 (𝑥 = (𝐺𝐶) → (dom 𝑥 = dom 𝑥 ↔ dom (𝐺𝐶) = dom (𝐺𝐶)))
14 rneq 5891 . . . . . . . . . . . . 13 (𝑥 = (𝐺𝐶) → ran 𝑥 = ran (𝐺𝐶))
15 df-ima 5646 . . . . . . . . . . . . 13 (𝐺𝐶) = ran (𝐺𝐶)
1614, 15eqtr4di 2794 . . . . . . . . . . . 12 (𝑥 = (𝐺𝐶) → ran 𝑥 = (𝐺𝐶))
1716unieqd 4879 . . . . . . . . . . 11 (𝑥 = (𝐺𝐶) → ran 𝑥 = (𝐺𝐶))
1817rneqd 5893 . . . . . . . . . 10 (𝑥 = (𝐺𝐶) → ran ran 𝑥 = ran (𝐺𝐶))
1918unieqd 4879 . . . . . . . . 9 (𝑥 = (𝐺𝐶) → ran ran 𝑥 = ran (𝐺𝐶))
20 suceq 6383 . . . . . . . . 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 7372 . . . . . . 7 (𝑥 = (𝐺𝐶) → (suc ran ran 𝑥 ·o (rank‘𝑦)) = (suc ran (𝐺𝐶) ·o (rank‘𝑦)))
23 fveq1 6841 . . . . . . . 8 (𝑥 = (𝐺𝐶) → (𝑥‘suc (rank‘𝑦)) = ((𝐺𝐶)‘suc (rank‘𝑦)))
2423fveq1d 6844 . . . . . . 7 (𝑥 = (𝐺𝐶) → ((𝑥‘suc (rank‘𝑦))‘𝑦) = (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦))
2522, 24oveq12d 7375 . . . . . 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 6849 . . . . . . . . . . . 12 (𝑥 = (𝐺𝐶) → (𝑥 dom 𝑥) = ((𝐺𝐶)‘ dom (𝐺𝐶)))
2827rneqd 5893 . . . . . . . . . . 11 (𝑥 = (𝐺𝐶) → ran (𝑥 dom 𝑥) = ran ((𝐺𝐶)‘ dom (𝐺𝐶)))
29 oieq2 9449 . . . . . . . . . . 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 5831 . . . . . . . . 9 (𝑥 = (𝐺𝐶) → OrdIso( E , ran (𝑥 dom 𝑥)) = OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))))
3231, 27coeq12d 5820 . . . . . . . 8 (𝑥 = (𝐺𝐶) → (OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) = (OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))))
3332imaeq1d 6012 . . . . . . 7 (𝑥 = (𝐺𝐶) → ((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦) = ((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦))
3433fveq2d 6846 . . . . . 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 4514 . . . . 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 5196 . . . 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 2736 . . . 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 6855 . . . . 5 (𝑅1‘dom (𝐺𝐶)) ∈ V
3938mptex 7173 . . . 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 6948 . . 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 7719 . . . . . . . 8 (𝐶 ∈ On → 𝐶 ⊆ On)
431, 42syl 17 . . . . . . 7 (𝜑𝐶 ⊆ On)
44 fnssres 6624 . . . . . . 7 ((𝐺 Fn On ∧ 𝐶 ⊆ On) → (𝐺𝐶) Fn 𝐶)
455, 43, 44sylancr 587 . . . . . 6 (𝜑 → (𝐺𝐶) Fn 𝐶)
4645fndmd 6607 . . . . 5 (𝜑 → dom (𝐺𝐶) = 𝐶)
4746fveq2d 6846 . . . 4 (𝜑 → (𝑅1‘dom (𝐺𝐶)) = (𝑅1𝐶))
4847mpteq1d 5200 . . 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)))))
4946adantr 481 . . . . . . 7 ((𝜑𝑦 ∈ (𝑅1𝐶)) → dom (𝐺𝐶) = 𝐶)
5049unieqd 4879 . . . . . . 7 ((𝜑𝑦 ∈ (𝑅1𝐶)) → dom (𝐺𝐶) = 𝐶)
5149, 50eqeq12d 2752 . . . . . 6 ((𝜑𝑦 ∈ (𝑅1𝐶)) → (dom (𝐺𝐶) = dom (𝐺𝐶) ↔ 𝐶 = 𝐶))
5251ifbid 4509 . . . . 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))))
53 rankr1ai 9734 . . . . . . . . . . . 12 (𝑦 ∈ (𝑅1𝐶) → (rank‘𝑦) ∈ 𝐶)
5453ad2antlr 725 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → (rank‘𝑦) ∈ 𝐶)
55 simpr 485 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → 𝐶 = 𝐶)
5654, 55eleqtrd 2840 . . . . . . . . . 10 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → (rank‘𝑦) ∈ 𝐶)
57 eloni 6327 . . . . . . . . . . . 12 (𝐶 ∈ On → Ord 𝐶)
58 ordsucuniel 7759 . . . . . . . . . . . 12 (Ord 𝐶 → ((rank‘𝑦) ∈ 𝐶 ↔ suc (rank‘𝑦) ∈ 𝐶))
591, 57, 583syl 18 . . . . . . . . . . 11 (𝜑 → ((rank‘𝑦) ∈ 𝐶 ↔ suc (rank‘𝑦) ∈ 𝐶))
6059ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → ((rank‘𝑦) ∈ 𝐶 ↔ suc (rank‘𝑦) ∈ 𝐶))
6156, 60mpbid 231 . . . . . . . . 9 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → suc (rank‘𝑦) ∈ 𝐶)
6261fvresd 6862 . . . . . . . 8 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → ((𝐺𝐶)‘suc (rank‘𝑦)) = (𝐺‘suc (rank‘𝑦)))
6362fveq1d 6844 . . . . . . 7 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑦))‘𝑦))
6463oveq2d 7373 . . . . . 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)))
6564ifeq1da 4517 . . . . 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))))
6650adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → dom (𝐺𝐶) = 𝐶)
6766fveq2d 6846 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → ((𝐺𝐶)‘ dom (𝐺𝐶)) = ((𝐺𝐶)‘ 𝐶))
681ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → 𝐶 ∈ On)
69 uniexg 7677 . . . . . . . . . . . . . . . . 17 (𝐶 ∈ On → 𝐶 ∈ V)
70 sucidg 6398 . . . . . . . . . . . . . . . . 17 ( 𝐶 ∈ V → 𝐶 ∈ suc 𝐶)
7168, 69, 703syl 18 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → 𝐶 ∈ suc 𝐶)
721adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (𝑅1𝐶)) → 𝐶 ∈ On)
73 orduniorsuc 7765 . . . . . . . . . . . . . . . . . 18 (Ord 𝐶 → (𝐶 = 𝐶𝐶 = suc 𝐶))
7472, 57, 733syl 18 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (𝑅1𝐶)) → (𝐶 = 𝐶𝐶 = suc 𝐶))
7574orcanai 1001 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → 𝐶 = suc 𝐶)
7671, 75eleqtrrd 2841 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → 𝐶𝐶)
7776fvresd 6862 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → ((𝐺𝐶)‘ 𝐶) = (𝐺 𝐶))
7867, 77eqtrd 2776 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → ((𝐺𝐶)‘ dom (𝐺𝐶)) = (𝐺 𝐶))
7978rneqd 5893 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → ran ((𝐺𝐶)‘ dom (𝐺𝐶)) = ran (𝐺 𝐶))
80 oieq2 9449 . . . . . . . . . . . 12 (ran ((𝐺𝐶)‘ dom (𝐺𝐶)) = ran (𝐺 𝐶) → OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) = OrdIso( E , ran (𝐺 𝐶)))
8179, 80syl 17 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) = OrdIso( E , ran (𝐺 𝐶)))
8281cnveqd 5831 . . . . . . . . . 10 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) = OrdIso( E , ran (𝐺 𝐶)))
8382, 78coeq12d 5820 . . . . . . . . 9 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → (OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) = (OrdIso( E , ran (𝐺 𝐶)) ∘ (𝐺 𝐶)))
84 dfac12.h . . . . . . . . 9 𝐻 = (OrdIso( E , ran (𝐺 𝐶)) ∘ (𝐺 𝐶))
8583, 84eqtr4di 2794 . . . . . . . 8 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → (OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) = 𝐻)
8685imaeq1d 6012 . . . . . . 7 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → ((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦) = (𝐻𝑦))
8786fveq2d 6846 . . . . . 6 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦)) = (𝐹‘(𝐻𝑦)))
8887ifeq2da 4518 . . . . 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))))
8952, 65, 883eqtrd 2780 . . . 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))))
9089mpteq2dva 5205 . . 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)))))
9148, 90eqtrd 2776 . 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)))))
924, 41, 913eqtrd 2780 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 205  wa 396  wo 845   = wceq 1541  wcel 2106  Vcvv 3445  wss 3910  ifcif 4486  𝒫 cpw 4560   cuni 4865  cmpt 5188   E cep 5536  ccnv 5632  dom cdm 5633  ran crn 5634  cres 5635  cima 5636  ccom 5637  Ord word 6316  Oncon0 6317  suc csuc 6319  Fun wfun 6490   Fn wfn 6491  1-1wf1 6493  cfv 6496  (class class class)co 7357  recscrecs 8316   +o coa 8409   ·o comu 8410  OrdIsocoi 9445  harchar 9492  𝑅1cr1 9698  rankcrnk 9699
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-se 5589  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-riota 7313  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-oi 9446  df-r1 9700  df-rank 9701
This theorem is referenced by:  dfac12lem2  10080
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