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Theorem dfac12lem1 10182
Description: Lemma for dfac12 10188. (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 8427 . . 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 8426 . . . . 5 𝐺 Fn On
6 fnfun 6659 . . . . 5 (𝐺 Fn On → Fun 𝐺)
75, 6ax-mp 5 . . . 4 Fun 𝐺
8 resfunexg 7231 . . . 4 ((Fun 𝐺𝐶 ∈ On) → (𝐺𝐶) ∈ V)
97, 1, 8sylancr 585 . . 3 (𝜑 → (𝐺𝐶) ∈ V)
10 dmeq 5909 . . . . . 6 (𝑥 = (𝐺𝐶) → dom 𝑥 = dom (𝐺𝐶))
1110fveq2d 6904 . . . . 5 (𝑥 = (𝐺𝐶) → (𝑅1‘dom 𝑥) = (𝑅1‘dom (𝐺𝐶)))
1210unieqd 4925 . . . . . . 7 (𝑥 = (𝐺𝐶) → dom 𝑥 = dom (𝐺𝐶))
1310, 12eqeq12d 2741 . . . . . 6 (𝑥 = (𝐺𝐶) → (dom 𝑥 = dom 𝑥 ↔ dom (𝐺𝐶) = dom (𝐺𝐶)))
14 rneq 5941 . . . . . . . . . . . . 13 (𝑥 = (𝐺𝐶) → ran 𝑥 = ran (𝐺𝐶))
15 df-ima 5694 . . . . . . . . . . . . 13 (𝐺𝐶) = ran (𝐺𝐶)
1614, 15eqtr4di 2783 . . . . . . . . . . . 12 (𝑥 = (𝐺𝐶) → ran 𝑥 = (𝐺𝐶))
1716unieqd 4925 . . . . . . . . . . 11 (𝑥 = (𝐺𝐶) → ran 𝑥 = (𝐺𝐶))
1817rneqd 5943 . . . . . . . . . 10 (𝑥 = (𝐺𝐶) → ran ran 𝑥 = ran (𝐺𝐶))
1918unieqd 4925 . . . . . . . . 9 (𝑥 = (𝐺𝐶) → ran ran 𝑥 = ran (𝐺𝐶))
20 suceq 6441 . . . . . . . . 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 7438 . . . . . . 7 (𝑥 = (𝐺𝐶) → (suc ran ran 𝑥 ·o (rank‘𝑦)) = (suc ran (𝐺𝐶) ·o (rank‘𝑦)))
23 fveq1 6899 . . . . . . . 8 (𝑥 = (𝐺𝐶) → (𝑥‘suc (rank‘𝑦)) = ((𝐺𝐶)‘suc (rank‘𝑦)))
2423fveq1d 6902 . . . . . . 7 (𝑥 = (𝐺𝐶) → ((𝑥‘suc (rank‘𝑦))‘𝑦) = (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦))
2522, 24oveq12d 7441 . . . . . 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 6907 . . . . . . . . . . . 12 (𝑥 = (𝐺𝐶) → (𝑥 dom 𝑥) = ((𝐺𝐶)‘ dom (𝐺𝐶)))
2827rneqd 5943 . . . . . . . . . . 11 (𝑥 = (𝐺𝐶) → ran (𝑥 dom 𝑥) = ran ((𝐺𝐶)‘ dom (𝐺𝐶)))
29 oieq2 9552 . . . . . . . . . . 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 5881 . . . . . . . . 9 (𝑥 = (𝐺𝐶) → OrdIso( E , ran (𝑥 dom 𝑥)) = OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))))
3231, 27coeq12d 5870 . . . . . . . 8 (𝑥 = (𝐺𝐶) → (OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) = (OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))))
3332imaeq1d 6067 . . . . . . 7 (𝑥 = (𝐺𝐶) → ((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦) = ((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦))
3433fveq2d 6904 . . . . . 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 4560 . . . . 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 5243 . . . 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 2725 . . . 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 6913 . . . . 5 (𝑅1‘dom (𝐺𝐶)) ∈ V
3938mptex 7239 . . . 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 7008 . . 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 7792 . . . . . . . 8 (𝐶 ∈ On → 𝐶 ⊆ On)
431, 42syl 17 . . . . . . 7 (𝜑𝐶 ⊆ On)
44 fnssres 6683 . . . . . . 7 ((𝐺 Fn On ∧ 𝐶 ⊆ On) → (𝐺𝐶) Fn 𝐶)
455, 43, 44sylancr 585 . . . . . 6 (𝜑 → (𝐺𝐶) Fn 𝐶)
4645fndmd 6664 . . . . 5 (𝜑 → dom (𝐺𝐶) = 𝐶)
4746fveq2d 6904 . . . 4 (𝜑 → (𝑅1‘dom (𝐺𝐶)) = (𝑅1𝐶))
4847mpteq1d 5247 . . 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 479 . . . . . . 7 ((𝜑𝑦 ∈ (𝑅1𝐶)) → dom (𝐺𝐶) = 𝐶)
5049unieqd 4925 . . . . . . 7 ((𝜑𝑦 ∈ (𝑅1𝐶)) → dom (𝐺𝐶) = 𝐶)
5149, 50eqeq12d 2741 . . . . . 6 ((𝜑𝑦 ∈ (𝑅1𝐶)) → (dom (𝐺𝐶) = dom (𝐺𝐶) ↔ 𝐶 = 𝐶))
5251ifbid 4555 . . . . 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 9837 . . . . . . . . . . . 12 (𝑦 ∈ (𝑅1𝐶) → (rank‘𝑦) ∈ 𝐶)
5453ad2antlr 725 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → (rank‘𝑦) ∈ 𝐶)
55 simpr 483 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → 𝐶 = 𝐶)
5654, 55eleqtrd 2827 . . . . . . . . . 10 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → (rank‘𝑦) ∈ 𝐶)
57 eloni 6385 . . . . . . . . . . . 12 (𝐶 ∈ On → Ord 𝐶)
58 ordsucuniel 7832 . . . . . . . . . . . 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 6920 . . . . . . . 8 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → ((𝐺𝐶)‘suc (rank‘𝑦)) = (𝐺‘suc (rank‘𝑦)))
6362fveq1d 6902 . . . . . . 7 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑦))‘𝑦))
6463oveq2d 7439 . . . . . 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 4563 . . . . 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 479 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → dom (𝐺𝐶) = 𝐶)
6766fveq2d 6904 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → ((𝐺𝐶)‘ dom (𝐺𝐶)) = ((𝐺𝐶)‘ 𝐶))
681ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → 𝐶 ∈ On)
69 uniexg 7750 . . . . . . . . . . . . . . . . 17 (𝐶 ∈ On → 𝐶 ∈ V)
70 sucidg 6456 . . . . . . . . . . . . . . . . 17 ( 𝐶 ∈ V → 𝐶 ∈ suc 𝐶)
7168, 69, 703syl 18 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → 𝐶 ∈ suc 𝐶)
721adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (𝑅1𝐶)) → 𝐶 ∈ On)
73 orduniorsuc 7838 . . . . . . . . . . . . . . . . . 18 (Ord 𝐶 → (𝐶 = 𝐶𝐶 = suc 𝐶))
7472, 57, 733syl 18 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (𝑅1𝐶)) → (𝐶 = 𝐶𝐶 = suc 𝐶))
7574orcanai 1000 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → 𝐶 = suc 𝐶)
7671, 75eleqtrrd 2828 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → 𝐶𝐶)
7776fvresd 6920 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → ((𝐺𝐶)‘ 𝐶) = (𝐺 𝐶))
7867, 77eqtrd 2765 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → ((𝐺𝐶)‘ dom (𝐺𝐶)) = (𝐺 𝐶))
7978rneqd 5943 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → ran ((𝐺𝐶)‘ dom (𝐺𝐶)) = ran (𝐺 𝐶))
80 oieq2 9552 . . . . . . . . . . . 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 5881 . . . . . . . . . 10 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) = OrdIso( E , ran (𝐺 𝐶)))
8382, 78coeq12d 5870 . . . . . . . . 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 2783 . . . . . . . 8 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → (OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) = 𝐻)
8685imaeq1d 6067 . . . . . . 7 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → ((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦) = (𝐻𝑦))
8786fveq2d 6904 . . . . . 6 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦)) = (𝐹‘(𝐻𝑦)))
8887ifeq2da 4564 . . . . 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 2769 . . . 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 5252 . . 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 2765 . 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 2769 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 394  wo 845   = wceq 1533  wcel 2098  Vcvv 3461  wss 3946  ifcif 4532  𝒫 cpw 4606   cuni 4912  cmpt 5235   E cep 5584  ccnv 5680  dom cdm 5681  ran crn 5682  cres 5683  cima 5684  ccom 5685  Ord word 6374  Oncon0 6375  suc csuc 6377  Fun wfun 6547   Fn wfn 6548  1-1wf1 6550  cfv 6553  (class class class)co 7423  recscrecs 8399   +o coa 8492   ·o comu 8493  OrdIsocoi 9548  harchar 9595  𝑅1cr1 9801  rankcrnk 9802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5579  df-eprel 5585  df-po 5593  df-so 5594  df-fr 5636  df-se 5637  df-we 5638  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-pred 6311  df-ord 6378  df-on 6379  df-lim 6380  df-suc 6381  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7379  df-ov 7426  df-om 7876  df-2nd 8003  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-oi 9549  df-r1 9803  df-rank 9804
This theorem is referenced by:  dfac12lem2  10183
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