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Theorem ctssdclemn0 7044
 Description: Lemma for ctssdc 7047. The ¬ ∅ ∈ 𝑆 case. (Contributed by Jim Kingdon, 16-Aug-2023.)
Hypotheses
Ref Expression
ctssdclemn0.ss (𝜑𝑆 ⊆ ω)
ctssdclemn0.dc (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑆)
ctssdclemn0.f (𝜑𝐹:𝑆onto𝐴)
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.
StepHypRef Expression
1 ctssdclemn0.f . . . . . . . . 9 (𝜑𝐹:𝑆onto𝐴)
21ad2antrr 480 . . . . . . . 8 (((𝜑𝑚 ∈ ω) ∧ 𝑚𝑆) → 𝐹:𝑆onto𝐴)
3 fof 5389 . . . . . . . 8 (𝐹:𝑆onto𝐴𝐹:𝑆𝐴)
42, 3syl 14 . . . . . . 7 (((𝜑𝑚 ∈ ω) ∧ 𝑚𝑆) → 𝐹:𝑆𝐴)
5 simpr 109 . . . . . . 7 (((𝜑𝑚 ∈ ω) ∧ 𝑚𝑆) → 𝑚𝑆)
64, 5ffvelrnd 5600 . . . . . 6 (((𝜑𝑚 ∈ ω) ∧ 𝑚𝑆) → (𝐹𝑚) ∈ 𝐴)
7 djulcl 6985 . . . . . 6 ((𝐹𝑚) ∈ 𝐴 → (inl‘(𝐹𝑚)) ∈ (𝐴 ⊔ 1o))
86, 7syl 14 . . . . 5 (((𝜑𝑚 ∈ ω) ∧ 𝑚𝑆) → (inl‘(𝐹𝑚)) ∈ (𝐴 ⊔ 1o))
9 0lt1o 6381 . . . . . . 7 ∅ ∈ 1o
10 djurcl 6986 . . . . . . 7 (∅ ∈ 1o → (inr‘∅) ∈ (𝐴 ⊔ 1o))
119, 10ax-mp 5 . . . . . 6 (inr‘∅) ∈ (𝐴 ⊔ 1o)
1211a1i 9 . . . . 5 (((𝜑𝑚 ∈ ω) ∧ ¬ 𝑚𝑆) → (inr‘∅) ∈ (𝐴 ⊔ 1o))
13 eleq1 2220 . . . . . . 7 (𝑛 = 𝑚 → (𝑛𝑆𝑚𝑆))
1413dcbid 824 . . . . . 6 (𝑛 = 𝑚 → (DECID 𝑛𝑆DECID 𝑚𝑆))
15 ctssdclemn0.dc . . . . . . 7 (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑆)
1615adantr 274 . . . . . 6 ((𝜑𝑚 ∈ ω) → ∀𝑛 ∈ ω DECID 𝑛𝑆)
17 simpr 109 . . . . . 6 ((𝜑𝑚 ∈ ω) → 𝑚 ∈ ω)
1814, 16, 17rspcdva 2821 . . . . 5 ((𝜑𝑚 ∈ ω) → DECID 𝑚𝑆)
198, 12, 18ifcldadc 3534 . . . 4 ((𝜑𝑚 ∈ ω) → if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)) ∈ (𝐴 ⊔ 1o))
2019fmpttd 5619 . . 3 (𝜑 → (𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅))):ω⟶(𝐴 ⊔ 1o))
211ad3antrrr 484 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) → 𝐹:𝑆onto𝐴)
22 simplr 520 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) → 𝑧𝐴)
23 foelrn 5698 . . . . . . . . 9 ((𝐹:𝑆onto𝐴𝑧𝐴) → ∃𝑦𝑆 𝑧 = (𝐹𝑦))
2421, 22, 23syl2anc 409 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) → ∃𝑦𝑆 𝑧 = (𝐹𝑦))
25 simplr 520 . . . . . . . . . . . 12 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → 𝑦𝑆)
2625iftrued 3512 . . . . . . . . . . 11 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → if(𝑦𝑆, (inl‘(𝐹𝑦)), (inr‘∅)) = (inl‘(𝐹𝑦)))
27 eqid 2157 . . . . . . . . . . . 12 (𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅))) = (𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))
28 eleq1 2220 . . . . . . . . . . . . 13 (𝑚 = 𝑦 → (𝑚𝑆𝑦𝑆))
29 2fveq3 5470 . . . . . . . . . . . . 13 (𝑚 = 𝑦 → (inl‘(𝐹𝑚)) = (inl‘(𝐹𝑦)))
3028, 29ifbieq1d 3527 . . . . . . . . . . . 12 (𝑚 = 𝑦 → if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)) = if(𝑦𝑆, (inl‘(𝐹𝑦)), (inr‘∅)))
31 ctssdclemn0.ss . . . . . . . . . . . . . 14 (𝜑𝑆 ⊆ ω)
3231ad5antr 488 . . . . . . . . . . . . 13 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → 𝑆 ⊆ ω)
3332, 25sseldd 3129 . . . . . . . . . . . 12 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → 𝑦 ∈ ω)
341, 3syl 14 . . . . . . . . . . . . . . . 16 (𝜑𝐹:𝑆𝐴)
3534ad5antr 488 . . . . . . . . . . . . . . 15 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → 𝐹:𝑆𝐴)
3635, 25ffvelrnd 5600 . . . . . . . . . . . . . 14 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → (𝐹𝑦) ∈ 𝐴)
37 djulcl 6985 . . . . . . . . . . . . . 14 ((𝐹𝑦) ∈ 𝐴 → (inl‘(𝐹𝑦)) ∈ (𝐴 ⊔ 1o))
3836, 37syl 14 . . . . . . . . . . . . 13 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → (inl‘(𝐹𝑦)) ∈ (𝐴 ⊔ 1o))
3926, 38eqeltrd 2234 . . . . . . . . . . . 12 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → if(𝑦𝑆, (inl‘(𝐹𝑦)), (inr‘∅)) ∈ (𝐴 ⊔ 1o))
4027, 30, 33, 39fvmptd3 5558 . . . . . . . . . . 11 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦) = if(𝑦𝑆, (inl‘(𝐹𝑦)), (inr‘∅)))
41 simpllr 524 . . . . . . . . . . . 12 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = (inl‘𝑧))
42 simpr 109 . . . . . . . . . . . . 13 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → 𝑧 = (𝐹𝑦))
4342fveq2d 5469 . . . . . . . . . . . 12 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → (inl‘𝑧) = (inl‘(𝐹𝑦)))
4441, 43eqtrd 2190 . . . . . . . . . . 11 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = (inl‘(𝐹𝑦)))
4526, 40, 443eqtr4rd 2201 . . . . . . . . . 10 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦))
4645ex 114 . . . . . . . . 9 (((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) → (𝑧 = (𝐹𝑦) → 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦)))
4746reximdva 2559 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) → (∃𝑦𝑆 𝑧 = (𝐹𝑦) → ∃𝑦𝑆 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦)))
4824, 47mpd 13 . . . . . . 7 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) → ∃𝑦𝑆 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦))
49 ssrexv 3193 . . . . . . . . 9 (𝑆 ⊆ ω → (∃𝑦𝑆 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦)))
5031, 49syl 14 . . . . . . . 8 (𝜑 → (∃𝑦𝑆 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦)))
5150ad3antrrr 484 . . . . . . 7 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) → (∃𝑦𝑆 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦)))
5248, 51mpd 13 . . . . . 6 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦))
5352rexlimdva2 2577 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) → (∃𝑧𝐴 𝑥 = (inl‘𝑧) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦)))
54 peano1 4551 . . . . . . . 8 ∅ ∈ ω
5554a1i 9 . . . . . . 7 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → ∅ ∈ ω)
56 ctssdclemn0.n0 . . . . . . . . . 10 (𝜑 → ¬ ∅ ∈ 𝑆)
5756ad3antrrr 484 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → ¬ ∅ ∈ 𝑆)
5857iffalsed 3515 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)) = (inr‘∅))
59 eleq1 2220 . . . . . . . . . 10 (𝑚 = ∅ → (𝑚𝑆 ↔ ∅ ∈ 𝑆))
60 2fveq3 5470 . . . . . . . . . 10 (𝑚 = ∅ → (inl‘(𝐹𝑚)) = (inl‘(𝐹‘∅)))
6159, 60ifbieq1d 3527 . . . . . . . . 9 (𝑚 = ∅ → if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)) = if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)))
6258, 11eqeltrdi 2248 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)) ∈ (𝐴 ⊔ 1o))
6327, 61, 55, 62fvmptd3 5558 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘∅) = if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)))
64 simpr 109 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑥 = (inr‘𝑧))
65 simplr 520 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑧 ∈ 1o)
66 el1o 6378 . . . . . . . . . . 11 (𝑧 ∈ 1o𝑧 = ∅)
6765, 66sylib 121 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑧 = ∅)
6867fveq2d 5469 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → (inr‘𝑧) = (inr‘∅))
6964, 68eqtrd 2190 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑥 = (inr‘∅))
7058, 63, 693eqtr4rd 2201 . . . . . . 7 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘∅))
71 fveq2 5465 . . . . . . . 8 (𝑦 = ∅ → ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦) = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘∅))
7271rspceeqv 2834 . . . . . . 7 ((∅ ∈ ω ∧ 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘∅)) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦))
7355, 70, 72syl2anc 409 . . . . . 6 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦))
7473rexlimdva2 2577 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) → (∃𝑧 ∈ 1o 𝑥 = (inr‘𝑧) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦)))
75 djur 7003 . . . . . . 7 (𝑥 ∈ (𝐴 ⊔ 1o) ↔ (∃𝑧𝐴 𝑥 = (inl‘𝑧) ∨ ∃𝑧 ∈ 1o 𝑥 = (inr‘𝑧)))
7675biimpi 119 . . . . . 6 (𝑥 ∈ (𝐴 ⊔ 1o) → (∃𝑧𝐴 𝑥 = (inl‘𝑧) ∨ ∃𝑧 ∈ 1o 𝑥 = (inr‘𝑧)))
7776adantl 275 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) → (∃𝑧𝐴 𝑥 = (inl‘𝑧) ∨ ∃𝑧 ∈ 1o 𝑥 = (inr‘𝑧)))
7853, 74, 77mpjaod 708 . . . 4 ((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦))
7978ralrimiva 2530 . . 3 (𝜑 → ∀𝑥 ∈ (𝐴 ⊔ 1o)∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦))
80 dffo3 5611 . . 3 ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅))):ω–onto→(𝐴 ⊔ 1o) ↔ ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅))):ω⟶(𝐴 ⊔ 1o) ∧ ∀𝑥 ∈ (𝐴 ⊔ 1o)∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦)))
8120, 79, 80sylanbrc 414 . 2 (𝜑 → (𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅))):ω–onto→(𝐴 ⊔ 1o))
82 omex 4550 . . . 4 ω ∈ V
8382mptex 5690 . . 3 (𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅))) ∈ V
84 foeq1 5385 . . 3 (𝑔 = (𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅))) → (𝑔:ω–onto→(𝐴 ⊔ 1o) ↔ (𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅))):ω–onto→(𝐴 ⊔ 1o)))
8583, 84spcev 2807 . 2 ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅))):ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
8681, 85syl 14 1 (𝜑 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 698  DECID wdc 820   = wceq 1335  ∃wex 1472   ∈ wcel 2128  ∀wral 2435  ∃wrex 2436   ⊆ wss 3102  ∅c0 3394  ifcif 3505   ↦ cmpt 4025  ωcom 4547  ⟶wf 5163  –onto→wfo 5165  ‘cfv 5167  1oc1o 6350   ⊔ cdju 6971  inlcinl 6979  inrcinr 6980 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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-iinf 4545 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4252  df-iord 4325  df-on 4327  df-suc 4330  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-1st 6082  df-2nd 6083  df-1o 6357  df-dju 6972  df-inl 6981  df-inr 6982 This theorem is referenced by:  ctssdc  7047
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