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Theorem ctssdclemn0 7103
Description: Lemma for ctssdc 7106. 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 5434 . . . . . . . 8 (𝐹:𝑆onto𝐴𝐹:𝑆𝐴)
42, 3syl 14 . . . . . . 7 (((𝜑𝑚 ∈ ω) ∧ 𝑚𝑆) → 𝐹:𝑆𝐴)
5 simpr 110 . . . . . . 7 (((𝜑𝑚 ∈ ω) ∧ 𝑚𝑆) → 𝑚𝑆)
64, 5ffvelcdmd 5648 . . . . . 6 (((𝜑𝑚 ∈ ω) ∧ 𝑚𝑆) → (𝐹𝑚) ∈ 𝐴)
7 djulcl 7044 . . . . . 6 ((𝐹𝑚) ∈ 𝐴 → (inl‘(𝐹𝑚)) ∈ (𝐴 ⊔ 1o))
86, 7syl 14 . . . . 5 (((𝜑𝑚 ∈ ω) ∧ 𝑚𝑆) → (inl‘(𝐹𝑚)) ∈ (𝐴 ⊔ 1o))
9 0lt1o 6435 . . . . . . 7 ∅ ∈ 1o
10 djurcl 7045 . . . . . . 7 (∅ ∈ 1o → (inr‘∅) ∈ (𝐴 ⊔ 1o))
119, 10ax-mp 5 . . . . . 6 (inr‘∅) ∈ (𝐴 ⊔ 1o)
1211a1i 9 . . . . 5 (((𝜑𝑚 ∈ ω) ∧ ¬ 𝑚𝑆) → (inr‘∅) ∈ (𝐴 ⊔ 1o))
13 eleq1 2240 . . . . . . 7 (𝑛 = 𝑚 → (𝑛𝑆𝑚𝑆))
1413dcbid 838 . . . . . 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 2846 . . . . 5 ((𝜑𝑚 ∈ ω) → DECID 𝑚𝑆)
198, 12, 18ifcldadc 3563 . . . 4 ((𝜑𝑚 ∈ ω) → if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)) ∈ (𝐴 ⊔ 1o))
2019fmpttd 5667 . . 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 5748 . . . . . . . . 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 3541 . . . . . . . . . . 11 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → if(𝑦𝑆, (inl‘(𝐹𝑦)), (inr‘∅)) = (inl‘(𝐹𝑦)))
27 eqid 2177 . . . . . . . . . . . 12 (𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅))) = (𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))
28 eleq1 2240 . . . . . . . . . . . . 13 (𝑚 = 𝑦 → (𝑚𝑆𝑦𝑆))
29 2fveq3 5516 . . . . . . . . . . . . 13 (𝑚 = 𝑦 → (inl‘(𝐹𝑚)) = (inl‘(𝐹𝑦)))
3028, 29ifbieq1d 3556 . . . . . . . . . . . 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 3156 . . . . . . . . . . . 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 5648 . . . . . . . . . . . . . 14 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → (𝐹𝑦) ∈ 𝐴)
37 djulcl 7044 . . . . . . . . . . . . . 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 2254 . . . . . . . . . . . 12 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → if(𝑦𝑆, (inl‘(𝐹𝑦)), (inr‘∅)) ∈ (𝐴 ⊔ 1o))
4027, 30, 33, 39fvmptd3 5605 . . . . . . . . . . 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 5515 . . . . . . . . . . . 12 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → (inl‘𝑧) = (inl‘(𝐹𝑦)))
4441, 43eqtrd 2210 . . . . . . . . . . 11 ((((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧𝐴) ∧ 𝑥 = (inl‘𝑧)) ∧ 𝑦𝑆) ∧ 𝑧 = (𝐹𝑦)) → 𝑥 = (inl‘(𝐹𝑦)))
4526, 40, 443eqtr4rd 2221 . . . . . . . . . 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 2579 . . . . . . . 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 3220 . . . . . . . . 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 2597 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) → (∃𝑧𝐴 𝑥 = (inl‘𝑧) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦)))
54 peano1 4590 . . . . . . . 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 3544 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)) = (inr‘∅))
59 eleq1 2240 . . . . . . . . . 10 (𝑚 = ∅ → (𝑚𝑆 ↔ ∅ ∈ 𝑆))
60 2fveq3 5516 . . . . . . . . . 10 (𝑚 = ∅ → (inl‘(𝐹𝑚)) = (inl‘(𝐹‘∅)))
6159, 60ifbieq1d 3556 . . . . . . . . 9 (𝑚 = ∅ → if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)) = if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)))
6258, 11eqeltrdi 2268 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → if(∅ ∈ 𝑆, (inl‘(𝐹‘∅)), (inr‘∅)) ∈ (𝐴 ⊔ 1o))
6327, 61, 55, 62fvmptd3 5605 . . . . . . . 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 6432 . . . . . . . . . . 11 (𝑧 ∈ 1o𝑧 = ∅)
6765, 66sylib 122 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑧 = ∅)
6867fveq2d 5515 . . . . . . . . 9 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → (inr‘𝑧) = (inr‘∅))
6964, 68eqtrd 2210 . . . . . . . 8 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑥 = (inr‘∅))
7058, 63, 693eqtr4rd 2221 . . . . . . 7 ((((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) ∧ 𝑧 ∈ 1o) ∧ 𝑥 = (inr‘𝑧)) → 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘∅))
71 fveq2 5511 . . . . . . . 8 (𝑦 = ∅ → ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦) = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘∅))
7271rspceeqv 2859 . . . . . . 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 2597 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) → (∃𝑧 ∈ 1o 𝑥 = (inr‘𝑧) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦)))
75 djur 7062 . . . . . . 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 718 . . . 4 ((𝜑𝑥 ∈ (𝐴 ⊔ 1o)) → ∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦))
7978ralrimiva 2550 . . 3 (𝜑 → ∀𝑥 ∈ (𝐴 ⊔ 1o)∃𝑦 ∈ ω 𝑥 = ((𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅)))‘𝑦))
80 dffo3 5659 . . 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 4589 . . . 4 ω ∈ V
8382mptex 5738 . . 3 (𝑚 ∈ ω ↦ if(𝑚𝑆, (inl‘(𝐹𝑚)), (inr‘∅))) ∈ V
84 foeq1 5430 . . 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 2832 . 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 708  DECID wdc 834   = wceq 1353  wex 1492  wcel 2148  wral 2455  wrex 2456  wss 3129  c0 3422  ifcif 3534  cmpt 4061  ωcom 4586  wf 5208  ontowfo 5210  cfv 5212  1oc1o 6404  cdju 7030  inlcinl 7038  inrcinr 7039
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-1st 6135  df-2nd 6136  df-1o 6411  df-dju 7031  df-inl 7040  df-inr 7041
This theorem is referenced by:  ctssdc  7106
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