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Theorem ctssdclemn0 7122
Description: Lemma for ctssdc 7125. 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 488 . . . . . . . 8 (((𝜑𝑚 ∈ ω) ∧ 𝑚𝑆) → 𝐹:𝑆onto𝐴)
3 fof 5450 . . . . . . . 8 (𝐹:𝑆onto𝐴𝐹:𝑆𝐴)
42, 3syl 14 . . . . . . 7 (((𝜑𝑚 ∈ ω) ∧ 𝑚𝑆) → 𝐹:𝑆𝐴)
5 simpr 110 . . . . . . 7 (((𝜑𝑚 ∈ ω) ∧ 𝑚𝑆) → 𝑚𝑆)
64, 5ffvelcdmd 5665 . . . . . 6 (((𝜑𝑚 ∈ ω) ∧ 𝑚𝑆) → (𝐹𝑚) ∈ 𝐴)
7 djulcl 7063 . . . . . 6 ((𝐹𝑚) ∈ 𝐴 → (inl‘(𝐹𝑚)) ∈ (𝐴 ⊔ 1o))
86, 7syl 14 . . . . 5 (((𝜑𝑚 ∈ ω) ∧ 𝑚𝑆) → (inl‘(𝐹𝑚)) ∈ (𝐴 ⊔ 1o))
9 0lt1o 6454 . . . . . . 7 ∅ ∈ 1o
10 djurcl 7064 . . . . . . 7 (∅ ∈ 1o → (inr‘∅) ∈ (𝐴 ⊔ 1o))
119, 10ax-mp 5 . . . . . 6 (inr‘∅) ∈ (𝐴 ⊔ 1o)
1211a1i 9 . . . . 5 (((𝜑𝑚 ∈ ω) ∧ ¬ 𝑚𝑆) → (inr‘∅) ∈ (𝐴 ⊔ 1o))
13 eleq1 2250 . . . . . . 7 (𝑛 = 𝑚 → (𝑛𝑆𝑚𝑆))
1413dcbid 839 . . . . . 6 (𝑛 = 𝑚 → (DECID 𝑛𝑆DECID 𝑚𝑆))
15 ctssdclemn0.dc . . . . . . 7 (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑆)
1615adantr 276 . . . . . 6 ((𝜑𝑚 ∈ ω) → ∀𝑛 ∈ ω DECID 𝑛𝑆)
17 simpr 110 . . . . . 6 ((𝜑𝑚 ∈ ω) → 𝑚 ∈ ω)
1814, 16, 17rspcdva 2858 . . . . 5 ((𝜑𝑚 ∈ ω) → DECID 𝑚𝑆)
198, 12, 18ifcldadc 3575 . . . 4 ((𝜑𝑚 ∈ ω) → if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)) ∈ (𝐴 ⊔ 1o))
2019fmpttd 5684 . . 3 (𝜑 → (𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅))):ω⟶(𝐴 ⊔ 1o))
211ad3antrrr 492 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) → 𝐹:𝑆onto𝐴)
22 simplr 528 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) → 𝑧𝐴)
23 foelrn 5766 . . . . . . . . 9 ((𝐹:𝑆onto𝐴𝑧𝐴) → ∃𝑦𝑆 𝑧 = (𝐹𝑦))
2421, 22, 23syl2anc 411 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) → ∃𝑦𝑆 𝑧 = (𝐹𝑦))
25 simplr 528 . . . . . . . . . . . 12 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → 𝑦𝑆)
2625iftrued 3553 . . . . . . . . . . 11 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → if(𝑦𝑆, (inl‘(𝐹𝑦)), (inr‘∅)) = (inl‘(𝐹𝑦)))
27 eqid 2187 . . . . . . . . . . . 12 (𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅))) = (𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))
28 eleq1 2250 . . . . . . . . . . . . 13 (𝑚 = 𝑦 → (𝑚𝑆𝑦𝑆))
29 2fveq3 5532 . . . . . . . . . . . . 13 (𝑚 = 𝑦 → (inl‘(𝐹𝑚)) = (inl‘(𝐹𝑦)))
3028, 29ifbieq1d 3568 . . . . . . . . . . . 12 (𝑚 = 𝑦 → if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)) = if(𝑦𝑆, (inl‘(𝐹𝑦)), (inr‘∅)))
31 ctssdclemn0.ss . . . . . . . . . . . . . 14 (𝜑𝑆 ⊆ ω)
3231ad5antr 496 . . . . . . . . . . . . 13 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → 𝑆 ⊆ ω)
3332, 25sseldd 3168 . . . . . . . . . . . 12 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → 𝑦 ∈ ω)
341, 3syl 14 . . . . . . . . . . . . . . . 16 (𝜑𝐹:𝑆𝐴)
3534ad5antr 496 . . . . . . . . . . . . . . 15 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → 𝐹:𝑆𝐴)
3635, 25ffvelcdmd 5665 . . . . . . . . . . . . . 14 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → (𝐹𝑦) ∈ 𝐴)
37 djulcl 7063 . . . . . . . . . . . . . 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 2264 . . . . . . . . . . . 12 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → if(𝑦𝑆, (inl‘(𝐹𝑦)), (inr‘∅)) ∈ (𝐴 ⊔ 1o))
4027, 30, 33, 39fvmptd3 5622 . . . . . . . . . . 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 534 . . . . . . . . . . . 12 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = (inl‘𝑧))
42 simpr 110 . . . . . . . . . . . . 13 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → 𝑧 = (𝐹𝑦))
4342fveq2d 5531 . . . . . . . . . . . 12 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → (inl‘𝑧) = (inl‘(𝐹𝑦)))
4441, 43eqtrd 2220 . . . . . . . . . . 11 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = (inl‘(𝐹𝑦)))
4526, 40, 443eqtr4rd 2231 . . . . . . . . . 10 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦))
4645ex 115 . . . . . . . . 9 (((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) → (𝑧 = (𝐹𝑦) → 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦)))
4746reximdva 2589 . . . . . . . 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 3232 . . . . . . . . 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 492 . . . . . . 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 2607 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) → (∃𝑧𝐴 𝑥 = (inl‘𝑧) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦)))
54 peano1 4605 . . . . . . . 8 ∅ ∈ ω
5554a1i 9 . . . . . . 7 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → ∅ ∈ ω)
56 ctssdclemn0.n0 . . . . . . . . . 10 (𝜑 → ¬ ∅ ∈ 𝑆)
5756ad3antrrr 492 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → ¬ ∅ ∈ 𝑆)
5857iffalsed 3556 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)) = (inr‘∅))
59 eleq1 2250 . . . . . . . . . 10 (𝑚 = ∅ → (𝑚𝑆 ↔ ∅ ∈ 𝑆))
60 2fveq3 5532 . . . . . . . . . 10 (𝑚 = ∅ → (inl‘(𝐹𝑚)) = (inl‘(𝐹‘∅)))
6159, 60ifbieq1d 3568 . . . . . . . . 9 (𝑚 = ∅ → if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)) = if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)))
6258, 11eqeltrdi 2278 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)) ∈ (𝐴 ⊔ 1o))
6327, 61, 55, 62fvmptd3 5622 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘∅) = if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)))
64 simpr 110 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑥 = (inr‘𝑧))
65 simplr 528 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑧 ∈ 1o)
66 el1o 6451 . . . . . . . . . . 11 (𝑧 ∈ 1o𝑧 = ∅)
6765, 66sylib 122 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑧 = ∅)
6867fveq2d 5531 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → (inr‘𝑧) = (inr‘∅))
6964, 68eqtrd 2220 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑥 = (inr‘∅))
7058, 63, 693eqtr4rd 2231 . . . . . . 7 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘∅))
71 fveq2 5527 . . . . . . . 8 (𝑦 = ∅ → ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦) = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘∅))
7271rspceeqv 2871 . . . . . . 7 ((∅ ∈ ω ∧ 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘∅)) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦))
7355, 70, 72syl2anc 411 . . . . . 6 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦))
7473rexlimdva2 2607 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) → (∃𝑧 ∈ 1o 𝑥 = (inr‘𝑧) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦)))
75 djur 7081 . . . . . . 7 (𝑥 ∈ (𝐴 ⊔ 1o) ↔ (∃𝑧𝐴 𝑥 = (inl‘𝑧) ∨ ∃𝑧 ∈ 1o 𝑥 = (inr‘𝑧)))
7675biimpi 120 . . . . . 6 (𝑥 ∈ (𝐴 ⊔ 1o) → (∃𝑧𝐴 𝑥 = (inl‘𝑧) ∨ ∃𝑧 ∈ 1o 𝑥 = (inr‘𝑧)))
7776adantl 277 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) → (∃𝑧𝐴 𝑥 = (inl‘𝑧) ∨ ∃𝑧 ∈ 1o 𝑥 = (inr‘𝑧)))
7853, 74, 77mpjaod 719 . . . 4 ((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦))
7978ralrimiva 2560 . . 3 (𝜑 → ∀𝑥 ∈ (𝐴 ⊔ 1o)∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦))
80 dffo3 5676 . . 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 417 . 2 (𝜑 → (𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅))):ω–onto→(𝐴 ⊔ 1o))
82 omex 4604 . . . 4 ω ∈ V
8382mptex 5755 . . 3 (𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅))) ∈ V
84 foeq1 5446 . . 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 2844 . 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 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  ontowfo 5226  cfv 5228  1oc1o 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
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