Theorem ctssdclemn0 6995

Description: Lemma for ctssdc 6998. 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 479	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝑚 ∈ 𝑆) → 𝐹:𝑆–onto→𝐴)
3		fof 5345	. . . . . . . 8 ⊢ (𝐹:𝑆–onto→𝐴 → 𝐹:𝑆⟶𝐴)
4	2, 3	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝑚 ∈ 𝑆) → 𝐹:𝑆⟶𝐴)
5		simpr 109	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝑚 ∈ 𝑆) → 𝑚 ∈ 𝑆)
6	4, 5	ffvelrnd 5556	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝑚 ∈ 𝑆) → (𝐹‘𝑚) ∈ 𝐴)
7		djulcl 6936	. . . . . 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 6337	. . . . . . 7 ⊢ ∅ ∈ 1o
10		djurcl 6937	. . . . . . 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 2202	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑛 ∈ 𝑆 ↔ 𝑚 ∈ 𝑆))
14	13	dcbid 823	. . . . . 6 ⊢ (𝑛 = 𝑚 → (DECID 𝑛 ∈ 𝑆 ↔ DECID 𝑚 ∈ 𝑆))
15		ctssdclemn0.dc	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)
16	15	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)
17		simpr 109	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → 𝑚 ∈ ω)
18	14, 16, 17	rspcdva 2794	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → DECID 𝑚 ∈ 𝑆)
19	8, 12, 18	ifcldadc 3501	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)) ∈ (𝐴 ⊔ 1o))
20	19	fmpttd 5575	. . 3 ⊢ (𝜑 → (𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅))):ω⟶(𝐴 ⊔ 1o))
21	1	ad3antrrr 483	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) → 𝐹:𝑆–onto→𝐴)
22		simplr 519	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) → 𝑧 ∈ 𝐴)
23		foelrn 5654	. . . . . . . . 9 ⊢ ((𝐹:𝑆–onto→𝐴 ∧ 𝑧 ∈ 𝐴) → ∃𝑦 ∈ 𝑆 𝑧 = (𝐹‘𝑦))
24	21, 22, 23	syl2anc 408	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) → ∃𝑦 ∈ 𝑆 𝑧 = (𝐹‘𝑦))
25		simplr 519	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑦 ∈ 𝑆)
26	25	iftrued 3481	. . . . . . . . . . 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 2139	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅))) = (𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))
28		eleq1 2202	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑦 → (𝑚 ∈ 𝑆 ↔ 𝑦 ∈ 𝑆))
29		2fveq3 5426	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑦 → (inl‘(𝐹‘𝑚)) = (inl‘(𝐹‘𝑦)))
30	28, 29	ifbieq1d 3494	. . . . . . . . . . . 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 487	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑆 ⊆ ω)
33	32, 25	sseldd 3098	. . . . . . . . . . . 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 487	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → 𝐹:𝑆⟶𝐴)
36	35, 25	ffvelrnd 5556	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → (𝐹‘𝑦) ∈ 𝐴)
37		djulcl 6936	. . . . . . . . . . . . . 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 2216	. . . . . . . . . . . 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 5514	. . . . . . . . . . 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 523	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = (inl‘𝑧))
42		simpr 109	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑧 = (𝐹‘𝑦))
43	42	fveq2d 5425	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → (inl‘𝑧) = (inl‘(𝐹‘𝑦)))
44	41, 43	eqtrd 2172	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = (inl‘(𝐹‘𝑦)))
45	26, 40, 44	3eqtr4rd 2183	. . . . . . . . . 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 114	. . . . . . . . 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 2534	. . . . . . . 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 3162	. . . . . . . . 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 483	. . . . . . 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 2552	. . . . 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 4508	. . . . . . . 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 483	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → ¬ ∅ ∈ 𝑆)
58	57	iffalsed 3484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)) = (inr‘∅))
59		eleq1 2202	. . . . . . . . . 10 ⊢ (𝑚 = ∅ → (𝑚 ∈ 𝑆 ↔ ∅ ∈ 𝑆))
60		2fveq3 5426	. . . . . . . . . 10 ⊢ (𝑚 = ∅ → (inl‘(𝐹‘𝑚)) = (inl‘(𝐹‘∅)))
61	59, 60	ifbieq1d 3494	. . . . . . . . 9 ⊢ (𝑚 = ∅ → if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)) = if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)))
62	58, 11	eqeltrdi 2230	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)) ∈ (𝐴 ⊔ 1o))
63	27, 61, 55, 62	fvmptd3 5514	. . . . . . . 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 109	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑥 = (inr‘𝑧))
65		simplr 519	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑧 ∈ 1o)
66		el1o 6334	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 1o ↔ 𝑧 = ∅)
67	65, 66	sylib 121	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑧 = ∅)
68	67	fveq2d 5425	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → (inr‘𝑧) = (inr‘∅))
69	64, 68	eqtrd 2172	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑥 = (inr‘∅))
70	58, 63, 69	3eqtr4rd 2183	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘∅))
71		fveq2 5421	. . . . . . . 8 ⊢ (𝑦 = ∅ → ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦) = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘∅))
72	71	rspceeqv 2807	. . . . . . 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 408	. . . . . 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 2552	. . . . 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 6954	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ⊔ 1o) ↔ (∃𝑧 ∈ 𝐴 𝑥 = (inl‘𝑧) ∨ ∃𝑧 ∈ 1o 𝑥 = (inr‘𝑧)))
76	75	biimpi 119	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ⊔ 1o) → (∃𝑧 ∈ 𝐴 𝑥 = (inl‘𝑧) ∨ ∃𝑧 ∈ 1o 𝑥 = (inr‘𝑧)))
77	76	adantl 275	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) → (∃𝑧 ∈ 𝐴 𝑥 = (inl‘𝑧) ∨ ∃𝑧 ∈ 1o 𝑥 = (inr‘𝑧)))
78	53, 74, 77	mpjaod 707	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦))
79	78	ralrimiva 2505	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ⊔ 1o)∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦))
80		dffo3 5567	. . 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 413	. 2 ⊢ (𝜑 → (𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅))):ω–onto→(𝐴 ⊔ 1o))
82		omex 4507	. . . 4 ⊢ ω ∈ V
83	82	mptex 5646	. . 3 ⊢ (𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅))) ∈ V
84		foeq1 5341	. . 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 2780	. 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 103   ∨ wo 697  DECID wdc 819   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417   ⊆ wss 3071  ∅c0 3363  ifcif 3474   ↦ cmpt 3989  ωcom 4504  ⟶wf 5119  –onto→wfo 5121  ‘cfv 5123  1_oc1o 6306   ⊔ cdju 6922  inlcinl 6930  inrcinr 6931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-1o 6313  df-dju 6923  df-inl 6932  df-inr 6933

This theorem is referenced by:  ctssdc  6998