Theorem dfac12lem1 10104

Description: Lemma for dfac12 10110. (Contributed by Mario Carneiro, 29-May-2015.)

Hypotheses

Ref	Expression
dfac12.1	⊢ (𝜑 → 𝐴 ∈ On)
dfac12.3	⊢ (𝜑 → 𝐹:𝒫 (har‘(𝑅1‘𝐴))–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	⊢ (𝜑 → 𝐶 ∈ On)
dfac12.h	⊢ 𝐻 = (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶))

Assertion

Ref	Expression
dfac12lem1	⊢ (𝜑 → (𝐺‘𝐶) = (𝑦 ∈ (𝑅1‘𝐶) ↦ if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦)))))

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

Step	Hyp	Ref	Expression
1		dfac12.5	. . 3 ⊢ (𝜑 → 𝐶 ∈ 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))))))
3	2	tfr2 8369	. . 3 ⊢ (𝐶 ∈ On → (𝐺‘𝐶) = ((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)))))‘(𝐺 ↾ 𝐶)))
4	1, 3	syl 17	. 2 ⊢ (𝜑 → (𝐺‘𝐶) = ((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)))))‘(𝐺 ↾ 𝐶)))
5	2	tfr1 8368	. . . . 5 ⊢ 𝐺 Fn On
6		fnfun 6621	. . . . 5 ⊢ (𝐺 Fn On → Fun 𝐺)
7	5, 6	ax-mp 5	. . . 4 ⊢ Fun 𝐺
8		resfunexg 7192	. . . 4 ⊢ ((Fun 𝐺 ∧ 𝐶 ∈ On) → (𝐺 ↾ 𝐶) ∈ V)
9	7, 1, 8	sylancr 587	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐶) ∈ V)
10		dmeq 5870	. . . . . 6 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → dom 𝑥 = dom (𝐺 ↾ 𝐶))
11	10	fveq2d 6865	. . . . 5 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → (𝑅1‘dom 𝑥) = (𝑅1‘dom (𝐺 ↾ 𝐶)))
12	10	unieqd 4887	. . . . . . 7 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ∪ dom 𝑥 = ∪ dom (𝐺 ↾ 𝐶))
13	10, 12	eqeq12d 2746	. . . . . 6 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → (dom 𝑥 = ∪ dom 𝑥 ↔ dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶)))
14		rneq 5903	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran 𝑥 = ran (𝐺 ↾ 𝐶))
15		df-ima 5654	. . . . . . . . . . . . 13 ⊢ (𝐺 “ 𝐶) = ran (𝐺 ↾ 𝐶)
16	14, 15	eqtr4di 2783	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran 𝑥 = (𝐺 “ 𝐶))
17	16	unieqd 4887	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ∪ ran 𝑥 = ∪ (𝐺 “ 𝐶))
18	17	rneqd 5905	. . . . . . . . . 10 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran ∪ ran 𝑥 = ran ∪ (𝐺 “ 𝐶))
19	18	unieqd 4887	. . . . . . . . 9 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ∪ ran ∪ ran 𝑥 = ∪ ran ∪ (𝐺 “ 𝐶))
20		suceq 6403	. . . . . . . . 9 ⊢ (∪ ran ∪ ran 𝑥 = ∪ ran ∪ (𝐺 “ 𝐶) → suc ∪ ran ∪ ran 𝑥 = suc ∪ ran ∪ (𝐺 “ 𝐶))
21	19, 20	syl 17	. . . . . . . 8 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → suc ∪ ran ∪ ran 𝑥 = suc ∪ ran ∪ (𝐺 “ 𝐶))
22	21	oveq1d 7405	. . . . . . 7 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → (suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑦)) = (suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)))
23		fveq1 6860	. . . . . . . 8 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → (𝑥‘suc (rank‘𝑦)) = ((𝐺 ↾ 𝐶)‘suc (rank‘𝑦)))
24	23	fveq1d 6863	. . . . . . 7 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ((𝑥‘suc (rank‘𝑦))‘𝑦) = (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦))
25	22, 24	oveq12d 7408	. . . . . 6 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ((suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑦)) +o ((𝑥‘suc (rank‘𝑦))‘𝑦)) = ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦)))
26		id 22	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → 𝑥 = (𝐺 ↾ 𝐶))
27	26, 12	fveq12d 6868	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → (𝑥‘∪ dom 𝑥) = ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶)))
28	27	rneqd 5905	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran (𝑥‘∪ dom 𝑥) = ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶)))
29		oieq2 9473	. . . . . . . . . . 11 ⊢ (ran (𝑥‘∪ dom 𝑥) = ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶)) → OrdIso( E , ran (𝑥‘∪ dom 𝑥)) = OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))))
30	28, 29	syl 17	. . . . . . . . . 10 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → OrdIso( E , ran (𝑥‘∪ dom 𝑥)) = OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))))
31	30	cnveqd 5842	. . . . . . . . 9 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) = ◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))))
32	31, 27	coeq12d 5831	. . . . . . . 8 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → (◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) = (◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))))
33	32	imaeq1d 6033	. . . . . . 7 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦) = ((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) “ 𝑦))
34	33	fveq2d 6865	. . . . . 6 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → (𝐹‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦)) = (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) “ 𝑦)))
35	13, 25, 34	ifbieq12d 4520	. . . . 5 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → 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 (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶), ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) “ 𝑦))))
36	11, 35	mpteq12dv 5197	. . . 4 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → (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 (𝐺 ↾ 𝐶)) ↦ if(dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶), ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) “ 𝑦)))))
37		eqid 2730	. . . 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 6874	. . . . 5 ⊢ (𝑅1‘dom (𝐺 ↾ 𝐶)) ∈ V
39	38	mptex 7200	. . . 4 ⊢ (𝑦 ∈ (𝑅1‘dom (𝐺 ↾ 𝐶)) ↦ if(dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶), ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) “ 𝑦)))) ∈ V
40	36, 37, 39	fvmpt 6971	. . 3 ⊢ ((𝐺 ↾ 𝐶) ∈ 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)))))‘(𝐺 ↾ 𝐶)) = (𝑦 ∈ (𝑅1‘dom (𝐺 ↾ 𝐶)) ↦ if(dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶), ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) “ 𝑦)))))
41	9, 40	syl 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)))))‘(𝐺 ↾ 𝐶)) = (𝑦 ∈ (𝑅1‘dom (𝐺 ↾ 𝐶)) ↦ if(dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶), ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) “ 𝑦)))))
42		onss 7764	. . . . . . . 8 ⊢ (𝐶 ∈ On → 𝐶 ⊆ On)
43	1, 42	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ On)
44		fnssres 6644	. . . . . . 7 ⊢ ((𝐺 Fn On ∧ 𝐶 ⊆ On) → (𝐺 ↾ 𝐶) Fn 𝐶)
45	5, 43, 44	sylancr 587	. . . . . 6 ⊢ (𝜑 → (𝐺 ↾ 𝐶) Fn 𝐶)
46	45	fndmd 6626	. . . . 5 ⊢ (𝜑 → dom (𝐺 ↾ 𝐶) = 𝐶)
47	46	fveq2d 6865	. . . 4 ⊢ (𝜑 → (𝑅1‘dom (𝐺 ↾ 𝐶)) = (𝑅1‘𝐶))
48	47	mpteq1d 5200	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝑅1‘dom (𝐺 ↾ 𝐶)) ↦ if(dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶), ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) “ 𝑦)))) = (𝑦 ∈ (𝑅1‘𝐶) ↦ if(dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶), ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) “ 𝑦)))))
49	46	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → dom (𝐺 ↾ 𝐶) = 𝐶)
50	49	unieqd 4887	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → ∪ dom (𝐺 ↾ 𝐶) = ∪ 𝐶)
51	49, 50	eqeq12d 2746	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → (dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶) ↔ 𝐶 = ∪ 𝐶))
52	51	ifbid 4515	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → if(dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶), ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) “ 𝑦))))
53		rankr1ai 9758	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑅1‘𝐶) → (rank‘𝑦) ∈ 𝐶)
54	53	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (rank‘𝑦) ∈ 𝐶)
55		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → 𝐶 = ∪ 𝐶)
56	54, 55	eleqtrd 2831	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (rank‘𝑦) ∈ ∪ 𝐶)
57		eloni 6345	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ On → Ord 𝐶)
58		ordsucuniel 7802	. . . . . . . . . . . 12 ⊢ (Ord 𝐶 → ((rank‘𝑦) ∈ ∪ 𝐶 ↔ suc (rank‘𝑦) ∈ 𝐶))
59	1, 57, 58	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((rank‘𝑦) ∈ ∪ 𝐶 ↔ suc (rank‘𝑦) ∈ 𝐶))
60	59	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((rank‘𝑦) ∈ ∪ 𝐶 ↔ suc (rank‘𝑦) ∈ 𝐶))
61	56, 60	mpbid 232	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → suc (rank‘𝑦) ∈ 𝐶)
62	61	fvresd 6881	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘suc (rank‘𝑦)) = (𝐺‘suc (rank‘𝑦)))
63	62	fveq1d 6863	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑦))‘𝑦))
64	63	oveq2d 7406	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦)) = ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)))
65	64	ifeq1da 4523	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) “ 𝑦))))
66	50	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ dom (𝐺 ↾ 𝐶) = ∪ 𝐶)
67	66	fveq2d 6865	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶)) = ((𝐺 ↾ 𝐶)‘∪ 𝐶))
68	1	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐶 ∈ On)
69		uniexg 7719	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 ∈ On → ∪ 𝐶 ∈ V)
70		sucidg 6418	. . . . . . . . . . . . . . . . 17 ⊢ (∪ 𝐶 ∈ V → ∪ 𝐶 ∈ suc ∪ 𝐶)
71	68, 69, 70	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶 ∈ suc ∪ 𝐶)
72	1	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → 𝐶 ∈ On)
73		orduniorsuc 7808	. . . . . . . . . . . . . . . . . 18 ⊢ (Ord 𝐶 → (𝐶 = ∪ 𝐶 ∨ 𝐶 = suc ∪ 𝐶))
74	72, 57, 73	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → (𝐶 = ∪ 𝐶 ∨ 𝐶 = suc ∪ 𝐶))
75	74	orcanai 1004	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐶 = suc ∪ 𝐶)
76	71, 75	eleqtrrd 2832	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶 ∈ 𝐶)
77	76	fvresd 6881	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘∪ 𝐶) = (𝐺‘∪ 𝐶))
78	67, 77	eqtrd 2765	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶)) = (𝐺‘∪ 𝐶))
79	78	rneqd 5905	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶)) = ran (𝐺‘∪ 𝐶))
80		oieq2 9473	. . . . . . . . . . . 12 ⊢ (ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶)) = ran (𝐺‘∪ 𝐶) → OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) = OrdIso( E , ran (𝐺‘∪ 𝐶)))
81	79, 80	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) = OrdIso( E , ran (𝐺‘∪ 𝐶)))
82	81	cnveqd 5842	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) = ◡OrdIso( E , ran (𝐺‘∪ 𝐶)))
83	82, 78	coeq12d 5831	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) = (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶)))
84		dfac12.h	. . . . . . . . 9 ⊢ 𝐻 = (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶))
85	83, 84	eqtr4di 2783	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) = 𝐻)
86	85	imaeq1d 6033	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) “ 𝑦) = (𝐻 “ 𝑦))
87	86	fveq2d 6865	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) “ 𝑦)) = (𝐹‘(𝐻 “ 𝑦)))
88	87	ifeq2da 4524	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))))
89	52, 65, 88	3eqtrd 2769	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → if(dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶), ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))))
90	89	mpteq2dva 5203	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝑅1‘𝐶) ↦ if(dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶), ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) “ 𝑦)))) = (𝑦 ∈ (𝑅1‘𝐶) ↦ if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦)))))
91	48, 90	eqtrd 2765	. 2 ⊢ (𝜑 → (𝑦 ∈ (𝑅1‘dom (𝐺 ↾ 𝐶)) ↦ if(dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶), ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) “ 𝑦)))) = (𝑦 ∈ (𝑅1‘𝐶) ↦ if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦)))))
92	4, 41, 91	3eqtrd 2769	1 ⊢ (𝜑 → (𝐺‘𝐶) = (𝑦 ∈ (𝑅1‘𝐶) ↦ if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)) +o ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ⊆ wss 3917  ifcif 4491  𝒫 cpw 4566  ∪ cuni 4874   ↦ cmpt 5191   E cep 5540  ^◡ccnv 5640  dom cdm 5641  ran crn 5642   ↾ cres 5643   “ cima 5644   ∘ ccom 5645  Ord word 6334  Oncon0 6335  suc csuc 6337  Fun wfun 6508   Fn wfn 6509  –1-1→wf1 6511  ‘cfv 6514  (class class class)co 7390  recscrecs 8342   +_o coa 8434   ·_o comu 8435  OrdIsocoi 9469  harchar 9516  𝑅₁cr1 9722  rankcrnk 9723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-oi 9470  df-r1 9724  df-rank 9725

This theorem is referenced by:  dfac12lem2  10105