Theorem ctmlemr 7169

Description: Lemma for ctm 7170. 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 6495	. . . . . . . . . 10 ⊢ ∅ ∈ 1o
2		djurcl 7113	. . . . . . . . . 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 5477	. . . . . . . . . . 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 4656	. . . . . . . . . . 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 5695	. . . . . . . . 9 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ ω) ∧ ¬ 𝑛 = ∅) → (𝑓‘∪ 𝑛) ∈ 𝐴)
11		djulcl 7112	. . . . . . . . 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 4651	. . . . . . . . 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 3587	. . . . . . 7 ⊢ (((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ ω) → if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))) ∈ (𝐴 ⊔ 1o))
16	15	fmpttd 5714	. . . . . 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 5796	. . . . . . . . . . 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 4628	. . . . . . . . . . . 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 4645	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ ω → Ord 𝑢)
26		ordtr 4410	. . . . . . . . . . . . . . . . . . 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 4446	. . . . . . . . . . . . . . . . . . 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 5559	. . . . . . . . . . . . . . . 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 2229	. . . . . . . . . . . . . . 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 5559	. . . . . . . . . . . . . 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 2229	. . . . . . . . . . . . 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 4629	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ ω → suc 𝑢 ≠ ∅)
37	36	neneqd 2385	. . . . . . . . . . . . . . 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 3568	. . . . . . . . . . . . 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 2229	. . . . . . . . . . . 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 2193	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛)))) = (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))
42		eqeq1 2200	. . . . . . . . . . . . . 14 ⊢ (𝑛 = suc 𝑢 → (𝑛 = ∅ ↔ suc 𝑢 = ∅))
43		unieq 3845	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = suc 𝑢 → ∪ 𝑛 = ∪ suc 𝑢)
44	43	fveq2d 5559	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = suc 𝑢 → (𝑓‘∪ 𝑛) = (𝑓‘∪ suc 𝑢))
45	44	fveq2d 5559	. . . . . . . . . . . . . 14 ⊢ (𝑛 = suc 𝑢 → (inl‘(𝑓‘∪ 𝑛)) = (inl‘(𝑓‘∪ suc 𝑢)))
46	42, 45	ifbieq2d 3582	. . . . . . . . . . . . 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 2271	. . . . . . . . . . . . 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 5652	. . . . . . . . . . . 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 2229	. . . . . . . . . . 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 5555	. . . . . . . . . . . 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 2883	. . . . . . . . . . 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 2616	. . . . . . . . 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 2612	. . . . . . . 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 4627	. . . . . . . . . 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 6492	. . . . . . . . . . . . . . 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 5559	. . . . . . . . . . . . 13 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o ∧ 𝑦 = (inr‘𝑤))) → (inr‘𝑤) = (inr‘∅))
62	57, 61	eqtrd 2226	. . . . . . . . . . . 12 ⊢ ((((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o ∧ 𝑦 = (inr‘𝑤))) → 𝑦 = (inr‘∅))
63		eqid 2193	. . . . . . . . . . . . 13 ⊢ ∅ = ∅
64	63	iftruei 3564	. . . . . . . . . . . 12 ⊢ if(∅ = ∅, (inr‘∅), (inl‘(𝑓‘∪ ∅))) = (inr‘∅)
65	62, 64	eqtr4di 2244	. . . . . . . . . . 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 2266	. . . . . . . . . . . . 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 2200	. . . . . . . . . . . . . 14 ⊢ (𝑛 = ∅ → (𝑛 = ∅ ↔ ∅ = ∅))
69		unieq 3845	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = ∅ → ∪ 𝑛 = ∪ ∅)
70	69	fveq2d 5559	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = ∅ → (𝑓‘∪ 𝑛) = (𝑓‘∪ ∅))
71	70	fveq2d 5559	. . . . . . . . . . . . . 14 ⊢ (𝑛 = ∅ → (inl‘(𝑓‘∪ 𝑛)) = (inl‘(𝑓‘∪ ∅)))
72	68, 71	ifbieq2d 3582	. . . . . . . . . . . . 13 ⊢ (𝑛 = ∅ → if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))) = if(∅ = ∅, (inr‘∅), (inl‘(𝑓‘∪ ∅))))
73	72, 41	fvmptg 5634	. . . . . . . . . . . 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 2229	. . . . . . . . . 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 5555	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘𝑧) = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛))))‘∅))
77	76	rspceeqv 2883	. . . . . . . . . 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 2612	. . . . . . . 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 7130	. . . . . . . . . 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 719	. . . . . . 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 2567	. . . . . 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 5706	. . . . . 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 4626	. . . . . . 7 ⊢ ω ∈ V
88	87	mptex 5785	. . . . . 6 ⊢ (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓‘∪ 𝑛)))) ∈ V
89		foeq1 5473	. . . . . 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 2856	. . . . 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 1830	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑓 𝑓:ω–onto→𝐴 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o)))
94		foeq1 5473	. . 3 ⊢ (𝑓 = 𝑔 → (𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ 𝑔:ω–onto→(𝐴 ⊔ 1o)))
95	94	cbvexv 1930	. 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 709  DECID wdc 835   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  ∅c0 3447  ifcif 3558  ∪ cuni 3836   ↦ cmpt 4091  Tr wtr 4128  Ord word 4394  suc csuc 4397  ωcom 4623  ⟶wf 5251  –onto→wfo 5253  ‘cfv 5255  1_oc1o 6464   ⊔ cdju 7098  inlcinl 7106  inrcinr 7107

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1st 6195  df-2nd 6196  df-1o 6471  df-dju 7099  df-inl 7108  df-inr 7109

This theorem is referenced by:  ctm  7170