Theorem dfac12lem1 9569

Description: Lemma for dfac12 9575. (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 8034	. . 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 8033	. . . . 5 ⊢ 𝐺 Fn On
6		fnfun 6453	. . . . 5 ⊢ (𝐺 Fn On → Fun 𝐺)
7	5, 6	ax-mp 5	. . . 4 ⊢ Fun 𝐺
8		resfunexg 6978	. . . 4 ⊢ ((Fun 𝐺 ∧ 𝐶 ∈ On) → (𝐺 ↾ 𝐶) ∈ V)
9	7, 1, 8	sylancr 589	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐶) ∈ V)
10		dmeq 5772	. . . . . 6 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → dom 𝑥 = dom (𝐺 ↾ 𝐶))
11	10	fveq2d 6674	. . . . 5 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → (𝑅1‘dom 𝑥) = (𝑅1‘dom (𝐺 ↾ 𝐶)))
12	10	unieqd 4852	. . . . . . 7 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ∪ dom 𝑥 = ∪ dom (𝐺 ↾ 𝐶))
13	10, 12	eqeq12d 2837	. . . . . 6 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → (dom 𝑥 = ∪ dom 𝑥 ↔ dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶)))
14		rneq 5806	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran 𝑥 = ran (𝐺 ↾ 𝐶))
15		df-ima 5568	. . . . . . . . . . . . 13 ⊢ (𝐺 “ 𝐶) = ran (𝐺 ↾ 𝐶)
16	14, 15	syl6eqr 2874	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran 𝑥 = (𝐺 “ 𝐶))
17	16	unieqd 4852	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ∪ ran 𝑥 = ∪ (𝐺 “ 𝐶))
18	17	rneqd 5808	. . . . . . . . . 10 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran ∪ ran 𝑥 = ran ∪ (𝐺 “ 𝐶))
19	18	unieqd 4852	. . . . . . . . 9 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ∪ ran ∪ ran 𝑥 = ∪ ran ∪ (𝐺 “ 𝐶))
20		suceq 6256	. . . . . . . . 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 7171	. . . . . . 7 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → (suc ∪ ran ∪ ran 𝑥 ·o (rank‘𝑦)) = (suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦)))
23		fveq1 6669	. . . . . . . 8 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → (𝑥‘suc (rank‘𝑦)) = ((𝐺 ↾ 𝐶)‘suc (rank‘𝑦)))
24	23	fveq1d 6672	. . . . . . 7 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ((𝑥‘suc (rank‘𝑦))‘𝑦) = (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦))
25	22, 24	oveq12d 7174	. . . . . 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 6677	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → (𝑥‘∪ dom 𝑥) = ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶)))
28	27	rneqd 5808	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran (𝑥‘∪ dom 𝑥) = ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶)))
29		oieq2 8977	. . . . . . . . . . 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 5746	. . . . . . . . 9 ⊢ (𝑥 = (𝐺 ↾ 𝐶) → ◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) = ◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))))
32	31, 27	coeq12d 5735	. . . . . . . 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 5928	. . . . . . 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 6674	. . . . . 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 4494	. . . . 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 5151	. . . 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 2821	. . . 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 6683	. . . . 5 ⊢ (𝑅1‘dom (𝐺 ↾ 𝐶)) ∈ V
39	38	mptex 6986	. . . 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 6768	. . 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 7505	. . . . . . . 8 ⊢ (𝐶 ∈ On → 𝐶 ⊆ On)
43	1, 42	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ On)
44		fnssres 6470	. . . . . . 7 ⊢ ((𝐺 Fn On ∧ 𝐶 ⊆ On) → (𝐺 ↾ 𝐶) Fn 𝐶)
45	5, 43, 44	sylancr 589	. . . . . 6 ⊢ (𝜑 → (𝐺 ↾ 𝐶) Fn 𝐶)
46		fndm 6455	. . . . . 6 ⊢ ((𝐺 ↾ 𝐶) Fn 𝐶 → dom (𝐺 ↾ 𝐶) = 𝐶)
47	45, 46	syl 17	. . . . 5 ⊢ (𝜑 → dom (𝐺 ↾ 𝐶) = 𝐶)
48	47	fveq2d 6674	. . . 4 ⊢ (𝜑 → (𝑅1‘dom (𝐺 ↾ 𝐶)) = (𝑅1‘𝐶))
49	48	mpteq1d 5155	. . 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)))))
50	47	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → dom (𝐺 ↾ 𝐶) = 𝐶)
51	50	unieqd 4852	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → ∪ dom (𝐺 ↾ 𝐶) = ∪ 𝐶)
52	50, 51	eqeq12d 2837	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → (dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶) ↔ 𝐶 = ∪ 𝐶))
53	52	ifbid 4489	. . . . 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))))
54		rankr1ai 9227	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑅1‘𝐶) → (rank‘𝑦) ∈ 𝐶)
55	54	ad2antlr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (rank‘𝑦) ∈ 𝐶)
56		simpr 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → 𝐶 = ∪ 𝐶)
57	55, 56	eleqtrd 2915	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (rank‘𝑦) ∈ ∪ 𝐶)
58		eloni 6201	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ On → Ord 𝐶)
59		ordsucuniel 7539	. . . . . . . . . . . 12 ⊢ (Ord 𝐶 → ((rank‘𝑦) ∈ ∪ 𝐶 ↔ suc (rank‘𝑦) ∈ 𝐶))
60	1, 58, 59	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((rank‘𝑦) ∈ ∪ 𝐶 ↔ suc (rank‘𝑦) ∈ 𝐶))
61	60	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((rank‘𝑦) ∈ ∪ 𝐶 ↔ suc (rank‘𝑦) ∈ 𝐶))
62	57, 61	mpbid 234	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → suc (rank‘𝑦) ∈ 𝐶)
63	62	fvresd 6690	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘suc (rank‘𝑦)) = (𝐺‘suc (rank‘𝑦)))
64	63	fveq1d 6672	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑦))‘𝑦))
65	64	oveq2d 7172	. . . . . 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)))
66	65	ifeq1da 4497	. . . . 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))))
67	51	adantr 483	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ dom (𝐺 ↾ 𝐶) = ∪ 𝐶)
68	67	fveq2d 6674	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶)) = ((𝐺 ↾ 𝐶)‘∪ 𝐶))
69	1	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐶 ∈ On)
70		uniexg 7466	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 ∈ On → ∪ 𝐶 ∈ V)
71		sucidg 6269	. . . . . . . . . . . . . . . . 17 ⊢ (∪ 𝐶 ∈ V → ∪ 𝐶 ∈ suc ∪ 𝐶)
72	69, 70, 71	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶 ∈ suc ∪ 𝐶)
73	1	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → 𝐶 ∈ On)
74		orduniorsuc 7545	. . . . . . . . . . . . . . . . . 18 ⊢ (Ord 𝐶 → (𝐶 = ∪ 𝐶 ∨ 𝐶 = suc ∪ 𝐶))
75	73, 58, 74	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → (𝐶 = ∪ 𝐶 ∨ 𝐶 = suc ∪ 𝐶))
76	75	orcanai 999	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐶 = suc ∪ 𝐶)
77	72, 76	eleqtrrd 2916	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶 ∈ 𝐶)
78	77	fvresd 6690	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘∪ 𝐶) = (𝐺‘∪ 𝐶))
79	68, 78	eqtrd 2856	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶)) = (𝐺‘∪ 𝐶))
80	79	rneqd 5808	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶)) = ran (𝐺‘∪ 𝐶))
81		oieq2 8977	. . . . . . . . . . . 12 ⊢ (ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶)) = ran (𝐺‘∪ 𝐶) → OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) = OrdIso( E , ran (𝐺‘∪ 𝐶)))
82	80, 81	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) = OrdIso( E , ran (𝐺‘∪ 𝐶)))
83	82	cnveqd 5746	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) = ◡OrdIso( E , ran (𝐺‘∪ 𝐶)))
84	83, 79	coeq12d 5735	. . . . . . . . 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)))
85		dfac12.h	. . . . . . . . 9 ⊢ 𝐻 = (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶))
86	84, 85	syl6eqr 2874	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) = 𝐻)
87	86	imaeq1d 5928	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶))) “ 𝑦) = (𝐻 “ 𝑦))
88	87	fveq2d 6674	. . . . . 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)))
89	88	ifeq2da 4498	. . . . 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))))
90	53, 66, 89	3eqtrd 2860	. . . 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))))
91	90	mpteq2dva 5161	. . 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)))))
92	49, 91	eqtrd 2856	. 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)))))
93	4, 41, 92	3eqtrd 2860	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 208   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114  Vcvv 3494   ⊆ wss 3936  ifcif 4467  𝒫 cpw 4539  ∪ cuni 4838   ↦ cmpt 5146   E cep 5464  ^◡ccnv 5554  dom cdm 5555  ran crn 5556   ↾ cres 5557   “ cima 5558   ∘ ccom 5559  Ord word 6190  Oncon0 6191  suc csuc 6193  Fun wfun 6349   Fn wfn 6350  –1-1→wf1 6352  ‘cfv 6355  (class class class)co 7156  recscrecs 8007   +_o coa 8099   ·_o comu 8100  OrdIsocoi 8973  harchar 9020  𝑅₁cr1 9191  rankcrnk 9192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-om 7581  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-oi 8974  df-r1 9193  df-rank 9194

This theorem is referenced by:  dfac12lem2  9570