Theorem ctssdclemn0 7075

Description: Lemma for ctssdc 7078. 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 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝑚 ∈ 𝑆) → 𝐹:𝑆–onto→𝐴)
3		fof 5410	. . . . . . . 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 5621	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ 𝑚 ∈ 𝑆) → (𝐹‘𝑚) ∈ 𝐴)
7		djulcl 7016	. . . . . 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 6408	. . . . . . 7 ⊢ ∅ ∈ 1o
10		djurcl 7017	. . . . . . 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 2229	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑛 ∈ 𝑆 ↔ 𝑚 ∈ 𝑆))
14	13	dcbid 828	. . . . . 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 2835	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → DECID 𝑚 ∈ 𝑆)
19	8, 12, 18	ifcldadc 3549	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)) ∈ (𝐴 ⊔ 1o))
20	19	fmpttd 5640	. . 3 ⊢ (𝜑 → (𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅))):ω⟶(𝐴 ⊔ 1o))
21	1	ad3antrrr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) → 𝐹:𝑆–onto→𝐴)
22		simplr 520	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) → 𝑧 ∈ 𝐴)
23		foelrn 5721	. . . . . . . . 9 ⊢ ((𝐹:𝑆–onto→𝐴 ∧ 𝑧 ∈ 𝐴) → ∃𝑦 ∈ 𝑆 𝑧 = (𝐹‘𝑦))
24	21, 22, 23	syl2anc 409	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) → ∃𝑦 ∈ 𝑆 𝑧 = (𝐹‘𝑦))
25		simplr 520	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑦 ∈ 𝑆)
26	25	iftrued 3527	. . . . . . . . . . 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 2165	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅))) = (𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))
28		eleq1 2229	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑦 → (𝑚 ∈ 𝑆 ↔ 𝑦 ∈ 𝑆))
29		2fveq3 5491	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑦 → (inl‘(𝐹‘𝑚)) = (inl‘(𝐹‘𝑦)))
30	28, 29	ifbieq1d 3542	. . . . . . . . . . . 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 488	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑆 ⊆ ω)
33	32, 25	sseldd 3143	. . . . . . . . . . . 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 488	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → 𝐹:𝑆⟶𝐴)
36	35, 25	ffvelrnd 5621	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → (𝐹‘𝑦) ∈ 𝐴)
37		djulcl 7016	. . . . . . . . . . . . . 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 2243	. . . . . . . . . . . 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 5579	. . . . . . . . . . 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 524	. . . . . . . . . . . 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 5490	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → (inl‘𝑧) = (inl‘(𝐹‘𝑦)))
44	41, 43	eqtrd 2198	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = (inl‘(𝐹‘𝑦)))
45	26, 40, 44	3eqtr4rd 2209	. . . . . . . . . 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 2568	. . . . . . . 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 3207	. . . . . . . . 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 484	. . . . . . 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 2586	. . . . 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 4571	. . . . . . . 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 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → ¬ ∅ ∈ 𝑆)
58	57	iffalsed 3530	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)) = (inr‘∅))
59		eleq1 2229	. . . . . . . . . 10 ⊢ (𝑚 = ∅ → (𝑚 ∈ 𝑆 ↔ ∅ ∈ 𝑆))
60		2fveq3 5491	. . . . . . . . . 10 ⊢ (𝑚 = ∅ → (inl‘(𝐹‘𝑚)) = (inl‘(𝐹‘∅)))
61	59, 60	ifbieq1d 3542	. . . . . . . . 9 ⊢ (𝑚 = ∅ → if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)) = if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)))
62	58, 11	eqeltrdi 2257	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)) ∈ (𝐴 ⊔ 1o))
63	27, 61, 55, 62	fvmptd3 5579	. . . . . . . 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 520	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑧 ∈ 1o)
66		el1o 6405	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 1o ↔ 𝑧 = ∅)
67	65, 66	sylib 121	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑧 = ∅)
68	67	fveq2d 5490	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → (inr‘𝑧) = (inr‘∅))
69	64, 68	eqtrd 2198	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑥 = (inr‘∅))
70	58, 63, 69	3eqtr4rd 2209	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘∅))
71		fveq2 5486	. . . . . . . 8 ⊢ (𝑦 = ∅ → ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦) = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘∅))
72	71	rspceeqv 2848	. . . . . . 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 409	. . . . . 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 2586	. . . . 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 7034	. . . . . . 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 708	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 1o)) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦))
79	78	ralrimiva 2539	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ⊔ 1o)∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅)))‘𝑦))
80		dffo3 5632	. . 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 414	. 2 ⊢ (𝜑 → (𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅))):ω–onto→(𝐴 ⊔ 1o))
82		omex 4570	. . . 4 ⊢ ω ∈ V
83	82	mptex 5711	. . 3 ⊢ (𝑚 ∈ ω ↦ if(𝑚 ∈ 𝑆, (inl‘(𝐹‘𝑚)), (inr‘∅))) ∈ V
84		foeq1 5406	. . 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 2821	. 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 698  DECID wdc 824   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445   ⊆ wss 3116  ∅c0 3409  ifcif 3520   ↦ cmpt 4043  ωcom 4567  ⟶wf 5184  –onto→wfo 5186  ‘cfv 5188  1_oc1o 6377   ⊔ cdju 7002  inlcinl 7010  inrcinr 7011

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109  df-1o 6384  df-dju 7003  df-inl 7012  df-inr 7013

This theorem is referenced by:  ctssdc  7078