Theorem ctmlemr 7263

Description: Lemma for ctm 7264. One of the directions of the biconditional. (Contributed by Jim Kingdon, 16-Mar-2023.)

Assertion

Ref	Expression
ctmlemr	⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑓 𝑓:ω–onto→𝐴 → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o)))

Distinct variable groups:   𝐴,𝑓   𝑥,𝑓

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem ctmlemr

Dummy variables 𝑔 𝑛 𝑢 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0lt1o 6576	. . . . . . . . . 10 ⊢ ∅ ∈ 1o
2		djurcl 7207	. . . . . . . . . 10 ⊢ (∅ ∈ 1o → (inr‘∅) ∈ (𝐴 ⊔ 1o))
3	1, 2	ax-mp 5	. . . . . . . . 9 ⊢ (inr‘∅) ∈ (𝐴 ⊔ 1o)
4	3	a1i 9	. . . . . . . 8 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ ω) ∧ 𝑛 = ∅) → (inr‘∅) ∈ (𝐴 ⊔ 1o))
5		simpllr 534	. . . . . . . . . . 11 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ ω) ∧ ¬ 𝑛 = ∅) → 𝑓:ω–onto→𝐴)
6		fof 5544	. . . . . . . . . . 11 ⊢ (𝑓:ω–onto→𝐴 → 𝑓:ω⟶𝐴)
7	5, 6	syl 14	. . . . . . . . . 10 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ ω) ∧ ¬ 𝑛 = ∅) → 𝑓:ω⟶𝐴)
8		nnpredcl 4712	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ω → ∪ 𝑛 ∈ ω)
9	8	ad2antlr 489	. . . . . . . . . 10 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ ω) ∧ ¬ 𝑛 = ∅) → ∪ 𝑛 ∈ ω)
10	7, 9	ffvelcdmd 5764	. . . . . . . . 9 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ ω) ∧ ¬ 𝑛 = ∅) → (𝑓‘∪ 𝑛) ∈ 𝐴)
11		djulcl 7206	. . . . . . . . 9 ⊢ ((𝑓‘∪ 𝑛) ∈ 𝐴 → (inl‘(𝑓‘∪ 𝑛)) ∈ (𝐴 ⊔ 1o))
12	10, 11	syl 14	. . . . . . . 8 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ ω) ∧ ¬ 𝑛 = ∅) → (inl‘(𝑓‘∪ 𝑛)) ∈ (𝐴 ⊔ 1o))
13		nndceq0 4707	. . . . . . . . 9 ⊢ (𝑛 ∈ ω → DECID 𝑛 = ∅)
14	13	adantl 277	. . . . . . . 8 ⊢ (((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ ω) → DECID 𝑛 = ∅)
15	4, 12, 14	ifcldadc 3632	. . . . . . 7 ⊢ (((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ ω) → if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))) ∈ (𝐴 ⊔ 1o))
16	15	fmpttd 5783	. . . . . 6 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) → (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛)))):ω⟶(𝐴 ⊔ 1o))
17		simpllr 534	. . . . . . . . . . 11 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) → 𝑓:ω–onto→𝐴)
18		simprl 529	. . . . . . . . . . 11 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) → 𝑤 ∈ 𝐴)
19		foelrn 5869	. . . . . . . . . . 11 ⊢ ((𝑓:ω–onto→𝐴 ∧ 𝑤 ∈ 𝐴) → ∃𝑢 ∈ ω 𝑤 = (𝑓‘𝑢))
20	17, 18, 19	syl2anc 411	. . . . . . . . . 10 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) → ∃𝑢 ∈ ω 𝑤 = (𝑓‘𝑢))
21		peano2 4684	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ω → suc 𝑢 ∈ ω)
22	21	ad2antrl 490	. . . . . . . . . . 11 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → suc 𝑢 ∈ ω)
23		simplrr 536	. . . . . . . . . . . . . 14 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → 𝑦 = (inl‘𝑤))
24		simprl 529	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → 𝑢 ∈ ω)
25		nnord 4701	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ ω → Ord 𝑢)
26		ordtr 4466	. . . . . . . . . . . . . . . . . . 19 ⊢ (Ord 𝑢 → Tr 𝑢)
27	24, 25, 26	3syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → Tr 𝑢)
28		unisucg 4502	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ ω → (Tr 𝑢 ↔ ∪ suc 𝑢 = 𝑢))
29	28	ad2antrl 490	. . . . . . . . . . . . . . . . . 18 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → (Tr 𝑢 ↔ ∪ suc 𝑢 = 𝑢))
30	27, 29	mpbid 147	. . . . . . . . . . . . . . . . 17 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → ∪ suc 𝑢 = 𝑢)
31	30	fveq2d 5627	. . . . . . . . . . . . . . . 16 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → (𝑓‘∪ suc 𝑢) = (𝑓‘𝑢))
32		simprr 531	. . . . . . . . . . . . . . . 16 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → 𝑤 = (𝑓‘𝑢))
33	31, 32	eqtr4d 2265	. . . . . . . . . . . . . . 15 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → (𝑓‘∪ suc 𝑢) = 𝑤)
34	33	fveq2d 5627	. . . . . . . . . . . . . 14 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → (inl‘(𝑓‘∪ suc 𝑢)) = (inl‘𝑤))
35	23, 34	eqtr4d 2265	. . . . . . . . . . . . 13 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → 𝑦 = (inl‘(𝑓‘∪ suc 𝑢)))
36		peano3 4685	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ ω → suc 𝑢 ≠ ∅)
37	36	neneqd 2421	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ ω → ¬ suc 𝑢 = ∅)
38	37	ad2antrl 490	. . . . . . . . . . . . . 14 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → ¬ suc 𝑢 = ∅)
39	38	iffalsed 3612	. . . . . . . . . . . . 13 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → if(suc 𝑢 = ∅, (inr‘∅), (inl‘(𝑓‘∪ suc 𝑢))) = (inl‘(𝑓‘∪ suc 𝑢)))
40	35, 39	eqtr4d 2265	. . . . . . . . . . . 12 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → 𝑦 = if(suc 𝑢 = ∅, (inr‘∅), (inl‘(𝑓‘∪ suc 𝑢))))
41		eqid 2229	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛)))) = (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))
42		eqeq1 2236	. . . . . . . . . . . . . 14 ⊢ (𝑛 = suc 𝑢 → (𝑛 = ∅ ↔ suc 𝑢 = ∅))
43		unieq 3896	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = suc 𝑢 → ∪ 𝑛 = ∪ suc 𝑢)
44	43	fveq2d 5627	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = suc 𝑢 → (𝑓‘∪ 𝑛) = (𝑓‘∪ suc 𝑢))
45	44	fveq2d 5627	. . . . . . . . . . . . . 14 ⊢ (𝑛 = suc 𝑢 → (inl‘(𝑓‘∪ 𝑛)) = (inl‘(𝑓‘∪ suc 𝑢)))
46	42, 45	ifbieq2d 3627	. . . . . . . . . . . . 13 ⊢ (𝑛 = suc 𝑢 → if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))) = if(suc 𝑢 = ∅, (inr‘∅), (inl‘(𝑓‘∪ suc 𝑢))))
47		simpllr 534	. . . . . . . . . . . . . 14 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → 𝑦 ∈ (𝐴 ⊔ 1o))
48	40, 47	eqeltrrd 2307	. . . . . . . . . . . . 13 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → if(suc 𝑢 = ∅, (inr‘∅), (inl‘(𝑓‘∪ suc 𝑢))) ∈ (𝐴 ⊔ 1o))
49	41, 46, 22, 48	fvmptd3 5721	. . . . . . . . . . . 12 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘suc 𝑢) = if(suc 𝑢 = ∅, (inr‘∅), (inl‘(𝑓‘∪ suc 𝑢))))
50	40, 49	eqtr4d 2265	. . . . . . . . . . 11 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘suc 𝑢))
51		fveq2 5623	. . . . . . . . . . . 12 ⊢ (𝑧 = suc 𝑢 → ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧) = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘suc 𝑢))
52	51	rspceeqv 2925	. . . . . . . . . . 11 ⊢ ((suc 𝑢 ∈ ω ∧ 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘suc 𝑢)) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧))
53	22, 50, 52	syl2anc 411	. . . . . . . . . 10 ⊢ (((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓‘𝑢))) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧))
54	20, 53	rexlimddv 2653	. . . . . . . . 9 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑦 = (inl‘𝑤))) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧))
55	54	rexlimdvaa 2649	. . . . . . . 8 ⊢ (((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) → (∃𝑤 ∈ 𝐴 𝑦 = (inl‘𝑤) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧)))
56		peano1 4683	. . . . . . . . . 10 ⊢ ∅ ∈ ω
57		simprr 531	. . . . . . . . . . . . 13 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o ∧ 𝑦 = (inr‘𝑤))) → 𝑦 = (inr‘𝑤))
58		simprl 529	. . . . . . . . . . . . . . 15 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o ∧ 𝑦 = (inr‘𝑤))) → 𝑤 ∈ 1o)
59		el1o 6573	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ 1o ↔ 𝑤 = ∅)
60	58, 59	sylib 122	. . . . . . . . . . . . . 14 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o ∧ 𝑦 = (inr‘𝑤))) → 𝑤 = ∅)
61	60	fveq2d 5627	. . . . . . . . . . . . 13 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o ∧ 𝑦 = (inr‘𝑤))) → (inr‘𝑤) = (inr‘∅))
62	57, 61	eqtrd 2262	. . . . . . . . . . . 12 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o ∧ 𝑦 = (inr‘𝑤))) → 𝑦 = (inr‘∅))
63		eqid 2229	. . . . . . . . . . . . 13 ⊢ ∅ = ∅
64	63	iftruei 3608	. . . . . . . . . . . 12 ⊢ if(∅ = ∅, (inr‘∅), (inl‘(𝑓‘∪ ∅))) = (inr‘∅)
65	62, 64	eqtr4di 2280	. . . . . . . . . . 11 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o ∧ 𝑦 = (inr‘𝑤))) → 𝑦 = if(∅ = ∅, (inr‘∅), (inl‘(𝑓‘∪ ∅))))
66	64, 3	eqeltri 2302	. . . . . . . . . . . . 13 ⊢ if(∅ = ∅, (inr‘∅), (inl‘(𝑓‘∪ ∅))) ∈ (𝐴 ⊔ 1o)
67	66	a1i 9	. . . . . . . . . . . 12 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o ∧ 𝑦 = (inr‘𝑤))) → if(∅ = ∅, (inr‘∅), (inl‘(𝑓‘∪ ∅))) ∈ (𝐴 ⊔ 1o))
68		eqeq1 2236	. . . . . . . . . . . . . 14 ⊢ (𝑛 = ∅ → (𝑛 = ∅ ↔ ∅ = ∅))
69		unieq 3896	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = ∅ → ∪ 𝑛 = ∪ ∅)
70	69	fveq2d 5627	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = ∅ → (𝑓‘∪ 𝑛) = (𝑓‘∪ ∅))
71	70	fveq2d 5627	. . . . . . . . . . . . . 14 ⊢ (𝑛 = ∅ → (inl‘(𝑓‘∪ 𝑛)) = (inl‘(𝑓‘∪ ∅)))
72	68, 71	ifbieq2d 3627	. . . . . . . . . . . . 13 ⊢ (𝑛 = ∅ → if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))) = if(∅ = ∅, (inr‘∅), (inl‘(𝑓‘∪ ∅))))
73	72, 41	fvmptg 5703	. . . . . . . . . . . 12 ⊢ ((∅ ∈ ω ∧ if(∅ = ∅, (inr‘∅), (inl‘(𝑓‘∪ ∅))) ∈ (𝐴 ⊔ 1o)) → ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘∅) = if(∅ = ∅, (inr‘∅), (inl‘(𝑓‘∪ ∅))))
74	56, 67, 73	sylancr 414	. . . . . . . . . . 11 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o ∧ 𝑦 = (inr‘𝑤))) → ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘∅) = if(∅ = ∅, (inr‘∅), (inl‘(𝑓‘∪ ∅))))
75	65, 74	eqtr4d 2265	. . . . . . . . . 10 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o ∧ 𝑦 = (inr‘𝑤))) → 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘∅))
76		fveq2 5623	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧) = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘∅))
77	76	rspceeqv 2925	. . . . . . . . . 10 ⊢ ((∅ ∈ ω ∧ 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘∅)) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧))
78	56, 75, 77	sylancr 414	. . . . . . . . 9 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o ∧ 𝑦 = (inr‘𝑤))) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧))
79	78	rexlimdvaa 2649	. . . . . . . 8 ⊢ (((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) → (∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧)))
80		djur 7224	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 ⊔ 1o) ↔ (∃𝑤 ∈ 𝐴 𝑦 = (inl‘𝑤) ∨ ∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤)))
81	80	biimpi 120	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐴 ⊔ 1o) → (∃𝑤 ∈ 𝐴 𝑦 = (inl‘𝑤) ∨ ∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤)))
82	81	adantl 277	. . . . . . . 8 ⊢ (((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) → (∃𝑤 ∈ 𝐴 𝑦 = (inl‘𝑤) ∨ ∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤)))
83	55, 79, 82	mpjaod 723	. . . . . . 7 ⊢ (((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧))
84	83	ralrimiva 2603	. . . . . 6 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) → ∀𝑦 ∈ (𝐴 ⊔ 1o)∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧))
85		dffo3 5775	. . . . . 6 ⊢ ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛)))):ω–onto→(𝐴 ⊔ 1o) ↔ ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛)))):ω⟶(𝐴 ⊔ 1o) ∧ ∀𝑦 ∈ (𝐴 ⊔ 1o)∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧)))
86	16, 84, 85	sylanbrc 417	. . . . 5 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) → (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛)))):ω–onto→(𝐴 ⊔ 1o))
87		omex 4682	. . . . . . 7 ⊢ ω ∈ V
88	87	mptex 5858	. . . . . 6 ⊢ (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛)))) ∈ V
89		foeq1 5540	. . . . . 6 ⊢ (𝑔 = (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛)))) → (𝑔:ω–onto→(𝐴 ⊔ 1o) ↔ (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛)))):ω–onto→(𝐴 ⊔ 1o)))
90	88, 89	spcev 2898	. . . . 5 ⊢ ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛)))):ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
91	86, 90	syl 14	. . . 4 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
92	91	ex 115	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑓:ω–onto→𝐴 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o)))
93	92	exlimdv 1865	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑓 𝑓:ω–onto→𝐴 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o)))
94		foeq1 5540	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ 𝑔:ω–onto→(𝐴 ⊔ 1o)))
95	94	cbvexv 1965	. 2 ⊢ (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
96	93, 95	imbitrrdi 162	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑓 𝑓:ω–onto→𝐴 → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713  DECID wdc 839   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  ∅c0 3491  ifcif 3602  ∪ cuni 3887   ↦ cmpt 4144  Tr wtr 4181  Ord word 4450  suc csuc 4453  ωcom 4679  ⟶wf 5310  –onto→wfo 5312  ‘cfv 5314  1_oc1o 6545   ⊔ cdju 7192  inlcinl 7200  inrcinr 7201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-iinf 4677

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-iord 4454  df-on 4456  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-1st 6276  df-2nd 6277  df-1o 6552  df-dju 7193  df-inl 7202  df-inr 7203

This theorem is referenced by:  ctm  7264