Theorem ctssdclemn0 7308

Description: Lemma for ctssdc 7311. 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 5559	. . . . . . . 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 5783	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝑚 ∈ 𝑆) → (𝐹‘𝑚) ∈ 𝐴)
7		djulcl 7249	. . . . . 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 6607	. . . . . . 7 ⊢ ∅ ∈ 1o
10		djurcl 7250	. . . . . . 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 2294	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑛 ∈ 𝑆 ↔ 𝑚 ∈ 𝑆))
14	13	dcbid 845	. . . . . 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 2915	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → DECID 𝑚 ∈ 𝑆)
19	8, 12, 18	ifcldadc 3635	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)) ∈ (𝐴 ⊔ 1o))
20	19	fmpttd 5802	. . 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 529	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) → 𝑧 ∈ 𝐴)
23		foelrn 5892	. . . . . . . . 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 529	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑦 ∈ 𝑆)
26	25	iftrued 3612	. . . . . . . . . . 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 2231	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅))) = (𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))
28		eleq1 2294	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑦 → (𝑚 ∈ 𝑆 ↔ 𝑦 ∈ 𝑆))
29		2fveq3 5644	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑦 → (inl‘(𝐹‘𝑚)) = (inl‘(𝐹‘𝑦)))
30	28, 29	ifbieq1d 3628	. . . . . . . . . . . 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 3228	. . . . . . . . . . . 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 5783	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → (𝐹‘𝑦) ∈ 𝐴)
37		djulcl 7249	. . . . . . . . . . . . . 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 2308	. . . . . . . . . . . 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 5740	. . . . . . . . . . 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 536	. . . . . . . . . . . 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 5643	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → (inl‘𝑧) = (inl‘(𝐹‘𝑦)))
44	41, 43	eqtrd 2264	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = (inl‘(𝐹‘𝑦)))
45	26, 40, 44	3eqtr4rd 2275	. . . . . . . . . 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 2634	. . . . . . . 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 3292	. . . . . . . . 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 2653	. . . . 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 4692	. . . . . . . 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 3615	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)) = (inr‘∅))
59		eleq1 2294	. . . . . . . . . 10 ⊢ (𝑚 = ∅ → (𝑚 ∈ 𝑆 ↔ ∅ ∈ 𝑆))
60		2fveq3 5644	. . . . . . . . . 10 ⊢ (𝑚 = ∅ → (inl‘(𝐹‘𝑚)) = (inl‘(𝐹‘∅)))
61	59, 60	ifbieq1d 3628	. . . . . . . . 9 ⊢ (𝑚 = ∅ → if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)) = if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)))
62	58, 11	eqeltrdi 2322	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)) ∈ (𝐴 ⊔ 1o))
63	27, 61, 55, 62	fvmptd3 5740	. . . . . . . 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 529	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑧 ∈ 1o)
66		el1o 6604	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 1o ↔ 𝑧 = ∅)
67	65, 66	sylib 122	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑧 = ∅)
68	67	fveq2d 5643	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → (inr‘𝑧) = (inr‘∅))
69	64, 68	eqtrd 2264	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑥 = (inr‘∅))
70	58, 63, 69	3eqtr4rd 2275	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘∅))
71		fveq2 5639	. . . . . . . 8 ⊢ (𝑦 = ∅ → ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦) = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘∅))
72	71	rspceeqv 2928	. . . . . . 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 2653	. . . . 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 7267	. . . . . . 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 725	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦))
79	78	ralrimiva 2605	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ⊔ 1o)∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦))
80		dffo3 5794	. . 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 4691	. . . 4 ⊢ ω ∈ V
83	82	mptex 5879	. . 3 ⊢ (𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅))) ∈ V
84		foeq1 5555	. . 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 2901	. 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 715  DECID wdc 841   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511   ⊆ wss 3200  ∅c0 3494  ifcif 3605   ↦ cmpt 4150  ωcom 4688  ⟶wf 5322  –onto→wfo 5324  ‘cfv 5326  1_oc1o 6574   ⊔ cdju 7235  inlcinl 7243  inrcinr 7244

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1st 6302  df-2nd 6303  df-1o 6581  df-dju 7236  df-inl 7245  df-inr 7246

This theorem is referenced by:  ctssdc  7311