Theorem ctssdclemn0 7122

Description: Lemma for ctssdc 7125. The ¬ ∅ ∈ 𝑆 case. (Contributed by Jim Kingdon, 16-Aug-2023.)

Hypotheses

Ref	Expression
ctssdclemn0.ss	⊢ (𝜑 → 𝑆 ⊆ ω)
ctssdclemn0.dc	⊢ (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)
ctssdclemn0.f	⊢ (𝜑 → 𝐹:𝑆–onto→𝐴)
ctssdclemn0.n0	⊢ (𝜑 → ¬ ∅ ∈ 𝑆)

Assertion

Ref	Expression
ctssdclemn0	⊢ (𝜑 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))

Distinct variable groups:   𝐴,𝑔   𝑔,𝐹   𝑆,𝑔   𝑆,𝑛

Allowed substitution hints:   𝜑(𝑔,𝑛)   𝐴(𝑛)   𝐹(𝑛)

Proof of Theorem ctssdclemn0

Dummy variables 𝑚 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ctssdclemn0.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑆–onto→𝐴)
2	1	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝑚 ∈ 𝑆) → 𝐹:𝑆–onto→𝐴)
3		fof 5450	. . . . . . . 8 ⊢ (𝐹:𝑆–onto→𝐴 → 𝐹:𝑆⟶𝐴)
4	2, 3	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝑚 ∈ 𝑆) → 𝐹:𝑆⟶𝐴)
5		simpr 110	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝑚 ∈ 𝑆) → 𝑚 ∈ 𝑆)
6	4, 5	ffvelcdmd 5665	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝑚 ∈ 𝑆) → (𝐹‘𝑚) ∈ 𝐴)
7		djulcl 7063	. . . . . 6 ⊢ ((𝐹‘𝑚) ∈ 𝐴 → (inl‘(𝐹‘𝑚)) ∈ (𝐴 ⊔ 1o))
8	6, 7	syl 14	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝑚 ∈ 𝑆) → (inl‘(𝐹‘𝑚)) ∈ (𝐴 ⊔ 1o))
9		0lt1o 6454	. . . . . . 7 ⊢ ∅ ∈ 1o
10		djurcl 7064	. . . . . . 7 ⊢ (∅ ∈ 1o → (inr‘∅) ∈ (𝐴 ⊔ 1o))
11	9, 10	ax-mp 5	. . . . . 6 ⊢ (inr‘∅) ∈ (𝐴 ⊔ 1o)
12	11	a1i 9	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ ¬ 𝑚 ∈ 𝑆) → (inr‘∅) ∈ (𝐴 ⊔ 1o))
13		eleq1 2250	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑛 ∈ 𝑆 ↔ 𝑚 ∈ 𝑆))
14	13	dcbid 839	. . . . . 6 ⊢ (𝑛 = 𝑚 → (DECID 𝑛 ∈ 𝑆 ↔ DECID 𝑚 ∈ 𝑆))
15		ctssdclemn0.dc	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)
16	15	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)
17		simpr 110	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → 𝑚 ∈ ω)
18	14, 16, 17	rspcdva 2858	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → DECID 𝑚 ∈ 𝑆)
19	8, 12, 18	ifcldadc 3575	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)) ∈ (𝐴 ⊔ 1o))
20	19	fmpttd 5684	. . 3 ⊢ (𝜑 → (𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅))):ω⟶(𝐴 ⊔ 1o))
21	1	ad3antrrr 492	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) → 𝐹:𝑆–onto→𝐴)
22		simplr 528	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) → 𝑧 ∈ 𝐴)
23		foelrn 5766	. . . . . . . . 9 ⊢ ((𝐹:𝑆–onto→𝐴 ∧ 𝑧 ∈ 𝐴) → ∃𝑦 ∈ 𝑆 𝑧 = (𝐹‘𝑦))
24	21, 22, 23	syl2anc 411	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) → ∃𝑦 ∈ 𝑆 𝑧 = (𝐹‘𝑦))
25		simplr 528	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑦 ∈ 𝑆)
26	25	iftrued 3553	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → if(𝑦 ∈ 𝑆, (inl‘(𝐹‘𝑦)), (inr‘∅)) = (inl‘(𝐹‘𝑦)))
27		eqid 2187	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅))) = (𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))
28		eleq1 2250	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑦 → (𝑚 ∈ 𝑆 ↔ 𝑦 ∈ 𝑆))
29		2fveq3 5532	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑦 → (inl‘(𝐹‘𝑚)) = (inl‘(𝐹‘𝑦)))
30	28, 29	ifbieq1d 3568	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑦 → if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)) = if(𝑦 ∈ 𝑆, (inl‘(𝐹‘𝑦)), (inr‘∅)))
31		ctssdclemn0.ss	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ⊆ ω)
32	31	ad5antr 496	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑆 ⊆ ω)
33	32, 25	sseldd 3168	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑦 ∈ ω)
34	1, 3	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:𝑆⟶𝐴)
35	34	ad5antr 496	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → 𝐹:𝑆⟶𝐴)
36	35, 25	ffvelcdmd 5665	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → (𝐹‘𝑦) ∈ 𝐴)
37		djulcl 7063	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑦) ∈ 𝐴 → (inl‘(𝐹‘𝑦)) ∈ (𝐴 ⊔ 1o))
38	36, 37	syl 14	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → (inl‘(𝐹‘𝑦)) ∈ (𝐴 ⊔ 1o))
39	26, 38	eqeltrd 2264	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → if(𝑦 ∈ 𝑆, (inl‘(𝐹‘𝑦)), (inr‘∅)) ∈ (𝐴 ⊔ 1o))
40	27, 30, 33, 39	fvmptd3 5622	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦) = if(𝑦 ∈ 𝑆, (inl‘(𝐹‘𝑦)), (inr‘∅)))
41		simpllr 534	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = (inl‘𝑧))
42		simpr 110	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑧 = (𝐹‘𝑦))
43	42	fveq2d 5531	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → (inl‘𝑧) = (inl‘(𝐹‘𝑦)))
44	41, 43	eqtrd 2220	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = (inl‘(𝐹‘𝑦)))
45	26, 40, 44	3eqtr4rd 2231	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦))
46	45	ex 115	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) → (𝑧 = (𝐹‘𝑦) → 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦)))
47	46	reximdva 2589	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) → (∃𝑦 ∈ 𝑆 𝑧 = (𝐹‘𝑦) → ∃𝑦 ∈ 𝑆 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦)))
48	24, 47	mpd 13	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) → ∃𝑦 ∈ 𝑆 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦))
49		ssrexv 3232	. . . . . . . . 9 ⊢ (𝑆 ⊆ ω → (∃𝑦 ∈ 𝑆 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦)))
50	31, 49	syl 14	. . . . . . . 8 ⊢ (𝜑 → (∃𝑦 ∈ 𝑆 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦)))
51	50	ad3antrrr 492	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) → (∃𝑦 ∈ 𝑆 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦)))
52	48, 51	mpd 13	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦))
53	52	rexlimdva2 2607	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) → (∃𝑧 ∈ 𝐴 𝑥 = (inl‘𝑧) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦)))
54		peano1 4605	. . . . . . . 8 ⊢ ∅ ∈ ω
55	54	a1i 9	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → ∅ ∈ ω)
56		ctssdclemn0.n0	. . . . . . . . . 10 ⊢ (𝜑 → ¬ ∅ ∈ 𝑆)
57	56	ad3antrrr 492	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → ¬ ∅ ∈ 𝑆)
58	57	iffalsed 3556	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)) = (inr‘∅))
59		eleq1 2250	. . . . . . . . . 10 ⊢ (𝑚 = ∅ → (𝑚 ∈ 𝑆 ↔ ∅ ∈ 𝑆))
60		2fveq3 5532	. . . . . . . . . 10 ⊢ (𝑚 = ∅ → (inl‘(𝐹‘𝑚)) = (inl‘(𝐹‘∅)))
61	59, 60	ifbieq1d 3568	. . . . . . . . 9 ⊢ (𝑚 = ∅ → if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)) = if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)))
62	58, 11	eqeltrdi 2278	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)) ∈ (𝐴 ⊔ 1o))
63	27, 61, 55, 62	fvmptd3 5622	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘∅) = if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)))
64		simpr 110	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑥 = (inr‘𝑧))
65		simplr 528	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑧 ∈ 1o)
66		el1o 6451	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 1o ↔ 𝑧 = ∅)
67	65, 66	sylib 122	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑧 = ∅)
68	67	fveq2d 5531	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → (inr‘𝑧) = (inr‘∅))
69	64, 68	eqtrd 2220	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑥 = (inr‘∅))
70	58, 63, 69	3eqtr4rd 2231	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘∅))
71		fveq2 5527	. . . . . . . 8 ⊢ (𝑦 = ∅ → ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦) = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘∅))
72	71	rspceeqv 2871	. . . . . . 7 ⊢ ((∅ ∈ ω ∧ 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘∅)) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦))
73	55, 70, 72	syl2anc 411	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦))
74	73	rexlimdva2 2607	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) → (∃𝑧 ∈ 1o 𝑥 = (inr‘𝑧) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦)))
75		djur 7081	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ⊔ 1o) ↔ (∃𝑧 ∈ 𝐴 𝑥 = (inl‘𝑧) ∨ ∃𝑧 ∈ 1o 𝑥 = (inr‘𝑧)))
76	75	biimpi 120	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ⊔ 1o) → (∃𝑧 ∈ 𝐴 𝑥 = (inl‘𝑧) ∨ ∃𝑧 ∈ 1o 𝑥 = (inr‘𝑧)))
77	76	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) → (∃𝑧 ∈ 𝐴 𝑥 = (inl‘𝑧) ∨ ∃𝑧 ∈ 1o 𝑥 = (inr‘𝑧)))
78	53, 74, 77	mpjaod 719	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦))
79	78	ralrimiva 2560	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ⊔ 1o)∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦))
80		dffo3 5676	. . 3 ⊢ ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅))):ω–onto→(𝐴 ⊔ 1o) ↔ ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅))):ω⟶(𝐴 ⊔ 1o) ∧ ∀𝑥 ∈ (𝐴 ⊔ 1o)∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦)))
81	20, 79, 80	sylanbrc 417	. 2 ⊢ (𝜑 → (𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅))):ω–onto→(𝐴 ⊔ 1o))
82		omex 4604	. . . 4 ⊢ ω ∈ V
83	82	mptex 5755	. . 3 ⊢ (𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅))) ∈ V
84		foeq1 5446	. . 3 ⊢ (𝑔 = (𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅))) → (𝑔:ω–onto→(𝐴 ⊔ 1o) ↔ (𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅))):ω–onto→(𝐴 ⊔ 1o)))
85	83, 84	spcev 2844	. 2 ⊢ ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅))):ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
86	81, 85	syl 14	1 ⊢ (𝜑 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709  DECID wdc 835   = wceq 1363  ∃wex 1502   ∈ wcel 2158  ∀wral 2465  ∃wrex 2466   ⊆ wss 3141  ∅c0 3434  ifcif 3546   ↦ cmpt 4076  ωcom 4601  ⟶wf 5224  –onto→wfo 5226  ‘cfv 5228  1_oc1o 6423   ⊔ cdju 7049  inlcinl 7057  inrcinr 7058

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-iinf 4599

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-1st 6154  df-2nd 6155  df-1o 6430  df-dju 7050  df-inl 7059  df-inr 7060

This theorem is referenced by:  ctssdc  7125