Theorem dfac12lem1 10182

Description: Lemma for dfac12 10188. (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 8437	. . 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 8436	. . . . 5 ⊢ 𝐺 Fn On
6		fnfun 6669	. . . . 5 ⊢ (𝐺 Fn On → Fun 𝐺)
7	5, 6	ax-mp 5	. . . 4 ⊢ Fun 𝐺
8		resfunexg 7235	. . . 4 ⊢ ((Fun 𝐺 ∧ 𝐶 ∈ On) → (𝐺 ↾ 𝐶) ∈ V)
9	7, 1, 8	sylancr 587	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐶) ∈ V)
10		dmeq 5917	. . . . . 6 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → dom 𝑥 = dom (𝐺 ↾ 𝐶))
11	10	fveq2d 6911	. . . . 5 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → (𝑅1‘dom 𝑥) = (𝑅1‘dom (𝐺 ↾ 𝐶)))
12	10	unieqd 4925	. . . . . . 7 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ∪ dom 𝑥 = ∪ dom (𝐺 ↾ 𝐶))
13	10, 12	eqeq12d 2751	. . . . . 6 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → (dom 𝑥 = ∪ dom 𝑥 ↔ dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶)))
14		rneq 5950	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran 𝑥 = ran (𝐺 ↾ 𝐶))
15		df-ima 5702	. . . . . . . . . . . . 13 ⊢ (𝐺 “ 𝐶) = ran (𝐺 ↾ 𝐶)
16	14, 15	eqtr4di 2793	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran 𝑥 = (𝐺 “ 𝐶))
17	16	unieqd 4925	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ∪ ran 𝑥 = ∪ (𝐺 “ 𝐶))
18	17	rneqd 5952	. . . . . . . . . 10 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran ∪ ran 𝑥 = ran ∪ (𝐺 “ 𝐶))
19	18	unieqd 4925	. . . . . . . . 9 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ∪ ran ∪ ran 𝑥 = ∪ ran ∪ (𝐺 “ 𝐶))
20		suceq 6452	. . . . . . . . 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 7446	. . . . . . 7 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → (suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑦)) = (suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)))
23		fveq1 6906	. . . . . . . 8 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → (𝑥‘suc (rank‘𝑦)) = ((𝐺 ↾ 𝐶)‘suc (rank‘𝑦)))
24	23	fveq1d 6909	. . . . . . 7 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ((𝑥‘suc (rank‘𝑦))‘𝑦) = (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦))
25	22, 24	oveq12d 7449	. . . . . 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 6914	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → (𝑥‘∪ dom 𝑥) = ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶)))
28	27	rneqd 5952	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran (𝑥‘∪ dom 𝑥) = ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶)))
29		oieq2 9551	. . . . . . . . . . 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 5889	. . . . . . . . 9 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) = ◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))))
32	31, 27	coeq12d 5878	. . . . . . . 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 6079	. . . . . . 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 6911	. . . . . 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 4559	. . . . 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 5239	. . . 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 2735	. . . 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 6920	. . . . 5 ⊢ (𝑅1‘dom (𝐺 ↾ 𝐶)) ∈ V
39	38	mptex 7243	. . . 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 7016	. . 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 7804	. . . . . . . 8 ⊢ (𝐶 ∈ On → 𝐶 ⊆ On)
43	1, 42	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ On)
44		fnssres 6692	. . . . . . 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 6674	. . . . 5 ⊢ (𝜑 → dom (𝐺 ↾ 𝐶) = 𝐶)
47	46	fveq2d 6911	. . . 4 ⊢ (𝜑 → (𝑅1‘dom (𝐺 ↾ 𝐶)) = (𝑅1‘𝐶))
48	47	mpteq1d 5243	. . 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 4925	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → ∪ dom (𝐺 ↾ 𝐶) = ∪ 𝐶)
51	49, 50	eqeq12d 2751	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → (dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶) ↔ 𝐶 = ∪ 𝐶))
52	51	ifbid 4554	. . . . 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 9836	. . . . . . . . . . . 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 2841	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (rank‘𝑦) ∈ ∪ 𝐶)
57		eloni 6396	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ On → Ord 𝐶)
58		ordsucuniel 7844	. . . . . . . . . . . 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 6927	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘suc (rank‘𝑦)) = (𝐺‘suc (rank‘𝑦)))
63	62	fveq1d 6909	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑦))‘𝑦))
64	63	oveq2d 7447	. . . . . 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 4562	. . . . 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 6911	. . . . . . . . . . . . . 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 7759	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 ∈ On → ∪ 𝐶 ∈ V)
70		sucidg 6467	. . . . . . . . . . . . . . . . 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 7850	. . . . . . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶 ∈ 𝐶)
77	76	fvresd 6927	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘∪ 𝐶) = (𝐺‘∪ 𝐶))
78	67, 77	eqtrd 2775	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶)) = (𝐺‘∪ 𝐶))
79	78	rneqd 5952	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶)) = ran (𝐺‘∪ 𝐶))
80		oieq2 9551	. . . . . . . . . . . 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 5889	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) = ◡OrdIso( E , ran (𝐺‘∪ 𝐶)))
83	82, 78	coeq12d 5878	. . . . . . . . 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 2793	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) = 𝐻)
86	85	imaeq1d 6079	. . . . . . 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 6911	. . . . . 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 4563	. . . . 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 2779	. . . 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 5248	. . 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 2775	. 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 2779	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 1537   ∈ wcel 2106  Vcvv 3478   ⊆ wss 3963  ifcif 4531  𝒫 cpw 4605  ∪ cuni 4912   ↦ cmpt 5231   E cep 5588  ^◡ccnv 5688  dom cdm 5689  ran crn 5690   ↾ cres 5691   “ cima 5692   ∘ ccom 5693  Ord word 6385  Oncon0 6386  suc csuc 6388  Fun wfun 6557   Fn wfn 6558  –1-1→wf1 6560  ‘cfv 6563  (class class class)co 7431  recscrecs 8409   +_o coa 8502   ·_o comu 8503  OrdIsocoi 9547  harchar 9594  𝑅₁cr1 9800  rankcrnk 9801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-oi 9548  df-r1 9802  df-rank 9803

This theorem is referenced by:  dfac12lem2  10183