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Theorem ctmlemr 7101
Description: Lemma for ctm 7102. One of the directions of the biconditional. (Contributed by Jim Kingdon, 16-Mar-2023.)
Assertion
Ref Expression
ctmlemr (∃𝑥 𝑥𝐴 → (∃𝑓 𝑓:ω–onto𝐴 → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o)))
Distinct variable groups:   𝐴,𝑓   𝑥,𝑓
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem ctmlemr
Dummy variables 𝑔 𝑛 𝑢 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0lt1o 6435 . . . . . . . . . 10 ∅ ∈ 1o
2 djurcl 7045 . . . . . . . . . 10 (∅ ∈ 1o → (inr‘∅) ∈ (𝐴 ⊔ 1o))
31, 2ax-mp 5 . . . . . . . . 9 (inr‘∅) ∈ (𝐴 ⊔ 1o)
43a1i 9 . . . . . . . 8 ((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑛 ∈ ω) ∧ 𝑛 = ∅) → (inr‘∅) ∈ (𝐴 ⊔ 1o))
5 simpllr 534 . . . . . . . . . . 11 ((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑛 ∈ ω) ∧ ¬ 𝑛 = ∅) → 𝑓:ω–onto𝐴)
6 fof 5434 . . . . . . . . . . 11 (𝑓:ω–onto𝐴𝑓:ω⟶𝐴)
75, 6syl 14 . . . . . . . . . 10 ((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑛 ∈ ω) ∧ ¬ 𝑛 = ∅) → 𝑓:ω⟶𝐴)
8 nnpredcl 4619 . . . . . . . . . . 11 (𝑛 ∈ ω → 𝑛 ∈ ω)
98ad2antlr 489 . . . . . . . . . 10 ((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑛 ∈ ω) ∧ ¬ 𝑛 = ∅) → 𝑛 ∈ ω)
107, 9ffvelcdmd 5648 . . . . . . . . 9 ((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑛 ∈ ω) ∧ ¬ 𝑛 = ∅) → (𝑓 𝑛) ∈ 𝐴)
11 djulcl 7044 . . . . . . . . 9 ((𝑓 𝑛) ∈ 𝐴 → (inl‘(𝑓 𝑛)) ∈ (𝐴 ⊔ 1o))
1210, 11syl 14 . . . . . . . 8 ((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑛 ∈ ω) ∧ ¬ 𝑛 = ∅) → (inl‘(𝑓 𝑛)) ∈ (𝐴 ⊔ 1o))
13 nndceq0 4614 . . . . . . . . 9 (𝑛 ∈ ω → DECID 𝑛 = ∅)
1413adantl 277 . . . . . . . 8 (((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑛 ∈ ω) → DECID 𝑛 = ∅)
154, 12, 14ifcldadc 3563 . . . . . . 7 (((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑛 ∈ ω) → if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))) ∈ (𝐴 ⊔ 1o))
1615fmpttd 5667 . . . . . 6 ((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) → (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛)))):ω⟶(𝐴 ⊔ 1o))
17 simpllr 534 . . . . . . . . . . 11 ((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) → 𝑓:ω–onto𝐴)
18 simprl 529 . . . . . . . . . . 11 ((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) → 𝑤𝐴)
19 foelrn 5748 . . . . . . . . . . 11 ((𝑓:ω–onto𝐴𝑤𝐴) → ∃𝑢 ∈ ω 𝑤 = (𝑓𝑢))
2017, 18, 19syl2anc 411 . . . . . . . . . 10 ((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) → ∃𝑢 ∈ ω 𝑤 = (𝑓𝑢))
21 peano2 4591 . . . . . . . . . . . 12 (𝑢 ∈ ω → suc 𝑢 ∈ ω)
2221ad2antrl 490 . . . . . . . . . . 11 (((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓𝑢))) → suc 𝑢 ∈ ω)
23 simplrr 536 . . . . . . . . . . . . . 14 (((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓𝑢))) → 𝑦 = (inl‘𝑤))
24 simprl 529 . . . . . . . . . . . . . . . . . . 19 (((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓𝑢))) → 𝑢 ∈ ω)
25 nnord 4608 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ ω → Ord 𝑢)
26 ordtr 4375 . . . . . . . . . . . . . . . . . . 19 (Ord 𝑢 → Tr 𝑢)
2724, 25, 263syl 17 . . . . . . . . . . . . . . . . . 18 (((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓𝑢))) → Tr 𝑢)
28 unisucg 4411 . . . . . . . . . . . . . . . . . . 19 (𝑢 ∈ ω → (Tr 𝑢 suc 𝑢 = 𝑢))
2928ad2antrl 490 . . . . . . . . . . . . . . . . . 18 (((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓𝑢))) → (Tr 𝑢 suc 𝑢 = 𝑢))
3027, 29mpbid 147 . . . . . . . . . . . . . . . . 17 (((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓𝑢))) → suc 𝑢 = 𝑢)
3130fveq2d 5515 . . . . . . . . . . . . . . . 16 (((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓𝑢))) → (𝑓 suc 𝑢) = (𝑓𝑢))
32 simprr 531 . . . . . . . . . . . . . . . 16 (((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓𝑢))) → 𝑤 = (𝑓𝑢))
3331, 32eqtr4d 2213 . . . . . . . . . . . . . . 15 (((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓𝑢))) → (𝑓 suc 𝑢) = 𝑤)
3433fveq2d 5515 . . . . . . . . . . . . . 14 (((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓𝑢))) → (inl‘(𝑓 suc 𝑢)) = (inl‘𝑤))
3523, 34eqtr4d 2213 . . . . . . . . . . . . 13 (((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓𝑢))) → 𝑦 = (inl‘(𝑓 suc 𝑢)))
36 peano3 4592 . . . . . . . . . . . . . . . 16 (𝑢 ∈ ω → suc 𝑢 ≠ ∅)
3736neneqd 2368 . . . . . . . . . . . . . . 15 (𝑢 ∈ ω → ¬ suc 𝑢 = ∅)
3837ad2antrl 490 . . . . . . . . . . . . . 14 (((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓𝑢))) → ¬ suc 𝑢 = ∅)
3938iffalsed 3544 . . . . . . . . . . . . 13 (((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓𝑢))) → if(suc 𝑢 = ∅, (inr‘∅), (inl‘(𝑓 suc 𝑢))) = (inl‘(𝑓 suc 𝑢)))
4035, 39eqtr4d 2213 . . . . . . . . . . . 12 (((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓𝑢))) → 𝑦 = if(suc 𝑢 = ∅, (inr‘∅), (inl‘(𝑓 suc 𝑢))))
41 eqid 2177 . . . . . . . . . . . . 13 (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛)))) = (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))))
42 eqeq1 2184 . . . . . . . . . . . . . 14 (𝑛 = suc 𝑢 → (𝑛 = ∅ ↔ suc 𝑢 = ∅))
43 unieq 3816 . . . . . . . . . . . . . . . 16 (𝑛 = suc 𝑢 𝑛 = suc 𝑢)
4443fveq2d 5515 . . . . . . . . . . . . . . 15 (𝑛 = suc 𝑢 → (𝑓 𝑛) = (𝑓 suc 𝑢))
4544fveq2d 5515 . . . . . . . . . . . . . 14 (𝑛 = suc 𝑢 → (inl‘(𝑓 𝑛)) = (inl‘(𝑓 suc 𝑢)))
4642, 45ifbieq2d 3558 . . . . . . . . . . . . 13 (𝑛 = suc 𝑢 → if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))) = if(suc 𝑢 = ∅, (inr‘∅), (inl‘(𝑓 suc 𝑢))))
47 simpllr 534 . . . . . . . . . . . . . 14 (((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓𝑢))) → 𝑦 ∈ (𝐴 ⊔ 1o))
4840, 47eqeltrrd 2255 . . . . . . . . . . . . 13 (((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓𝑢))) → if(suc 𝑢 = ∅, (inr‘∅), (inl‘(𝑓 suc 𝑢))) ∈ (𝐴 ⊔ 1o))
4941, 46, 22, 48fvmptd3 5605 . . . . . . . . . . . 12 (((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓𝑢))) → ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))))‘suc 𝑢) = if(suc 𝑢 = ∅, (inr‘∅), (inl‘(𝑓 suc 𝑢))))
5040, 49eqtr4d 2213 . . . . . . . . . . 11 (((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓𝑢))) → 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))))‘suc 𝑢))
51 fveq2 5511 . . . . . . . . . . . 12 (𝑧 = suc 𝑢 → ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))))‘𝑧) = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))))‘suc 𝑢))
5251rspceeqv 2859 . . . . . . . . . . 11 ((suc 𝑢 ∈ ω ∧ 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))))‘suc 𝑢)) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))))‘𝑧))
5322, 50, 52syl2anc 411 . . . . . . . . . 10 (((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) ∧ (𝑢 ∈ ω ∧ 𝑤 = (𝑓𝑢))) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))))‘𝑧))
5420, 53rexlimddv 2599 . . . . . . . . 9 ((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤𝐴𝑦 = (inl‘𝑤))) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))))‘𝑧))
5554rexlimdvaa 2595 . . . . . . . 8 (((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) → (∃𝑤𝐴 𝑦 = (inl‘𝑤) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))))‘𝑧)))
56 peano1 4590 . . . . . . . . . 10 ∅ ∈ ω
57 simprr 531 . . . . . . . . . . . . 13 ((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o𝑦 = (inr‘𝑤))) → 𝑦 = (inr‘𝑤))
58 simprl 529 . . . . . . . . . . . . . . 15 ((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o𝑦 = (inr‘𝑤))) → 𝑤 ∈ 1o)
59 el1o 6432 . . . . . . . . . . . . . . 15 (𝑤 ∈ 1o𝑤 = ∅)
6058, 59sylib 122 . . . . . . . . . . . . . 14 ((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o𝑦 = (inr‘𝑤))) → 𝑤 = ∅)
6160fveq2d 5515 . . . . . . . . . . . . 13 ((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o𝑦 = (inr‘𝑤))) → (inr‘𝑤) = (inr‘∅))
6257, 61eqtrd 2210 . . . . . . . . . . . 12 ((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o𝑦 = (inr‘𝑤))) → 𝑦 = (inr‘∅))
63 eqid 2177 . . . . . . . . . . . . 13 ∅ = ∅
6463iftruei 3540 . . . . . . . . . . . 12 if(∅ = ∅, (inr‘∅), (inl‘(𝑓 ∅))) = (inr‘∅)
6562, 64eqtr4di 2228 . . . . . . . . . . 11 ((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o𝑦 = (inr‘𝑤))) → 𝑦 = if(∅ = ∅, (inr‘∅), (inl‘(𝑓 ∅))))
6664, 3eqeltri 2250 . . . . . . . . . . . . 13 if(∅ = ∅, (inr‘∅), (inl‘(𝑓 ∅))) ∈ (𝐴 ⊔ 1o)
6766a1i 9 . . . . . . . . . . . 12 ((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o𝑦 = (inr‘𝑤))) → if(∅ = ∅, (inr‘∅), (inl‘(𝑓 ∅))) ∈ (𝐴 ⊔ 1o))
68 eqeq1 2184 . . . . . . . . . . . . . 14 (𝑛 = ∅ → (𝑛 = ∅ ↔ ∅ = ∅))
69 unieq 3816 . . . . . . . . . . . . . . . 16 (𝑛 = ∅ → 𝑛 = ∅)
7069fveq2d 5515 . . . . . . . . . . . . . . 15 (𝑛 = ∅ → (𝑓 𝑛) = (𝑓 ∅))
7170fveq2d 5515 . . . . . . . . . . . . . 14 (𝑛 = ∅ → (inl‘(𝑓 𝑛)) = (inl‘(𝑓 ∅)))
7268, 71ifbieq2d 3558 . . . . . . . . . . . . 13 (𝑛 = ∅ → if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))) = if(∅ = ∅, (inr‘∅), (inl‘(𝑓 ∅))))
7372, 41fvmptg 5588 . . . . . . . . . . . 12 ((∅ ∈ ω ∧ if(∅ = ∅, (inr‘∅), (inl‘(𝑓 ∅))) ∈ (𝐴 ⊔ 1o)) → ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))))‘∅) = if(∅ = ∅, (inr‘∅), (inl‘(𝑓 ∅))))
7456, 67, 73sylancr 414 . . . . . . . . . . 11 ((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o𝑦 = (inr‘𝑤))) → ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))))‘∅) = if(∅ = ∅, (inr‘∅), (inl‘(𝑓 ∅))))
7565, 74eqtr4d 2213 . . . . . . . . . 10 ((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o𝑦 = (inr‘𝑤))) → 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))))‘∅))
76 fveq2 5511 . . . . . . . . . . 11 (𝑧 = ∅ → ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))))‘𝑧) = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))))‘∅))
7776rspceeqv 2859 . . . . . . . . . 10 ((∅ ∈ ω ∧ 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))))‘∅)) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))))‘𝑧))
7856, 75, 77sylancr 414 . . . . . . . . 9 ((((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) ∧ (𝑤 ∈ 1o𝑦 = (inr‘𝑤))) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))))‘𝑧))
7978rexlimdvaa 2595 . . . . . . . 8 (((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) → (∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))))‘𝑧)))
80 djur 7062 . . . . . . . . . 10 (𝑦 ∈ (𝐴 ⊔ 1o) ↔ (∃𝑤𝐴 𝑦 = (inl‘𝑤) ∨ ∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤)))
8180biimpi 120 . . . . . . . . 9 (𝑦 ∈ (𝐴 ⊔ 1o) → (∃𝑤𝐴 𝑦 = (inl‘𝑤) ∨ ∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤)))
8281adantl 277 . . . . . . . 8 (((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) → (∃𝑤𝐴 𝑦 = (inl‘𝑤) ∨ ∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤)))
8355, 79, 82mpjaod 718 . . . . . . 7 (((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) ∧ 𝑦 ∈ (𝐴 ⊔ 1o)) → ∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))))‘𝑧))
8483ralrimiva 2550 . . . . . 6 ((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) → ∀𝑦 ∈ (𝐴 ⊔ 1o)∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))))‘𝑧))
85 dffo3 5659 . . . . . 6 ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛)))):ω–onto→(𝐴 ⊔ 1o) ↔ ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛)))):ω⟶(𝐴 ⊔ 1o) ∧ ∀𝑦 ∈ (𝐴 ⊔ 1o)∃𝑧 ∈ ω 𝑦 = ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛))))‘𝑧)))
8616, 84, 85sylanbrc 417 . . . . 5 ((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) → (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛)))):ω–onto→(𝐴 ⊔ 1o))
87 omex 4589 . . . . . . 7 ω ∈ V
8887mptex 5738 . . . . . 6 (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛)))) ∈ V
89 foeq1 5430 . . . . . 6 (𝑔 = (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛)))) → (𝑔:ω–onto→(𝐴 ⊔ 1o) ↔ (𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛)))):ω–onto→(𝐴 ⊔ 1o)))
9088, 89spcev 2832 . . . . 5 ((𝑛 ∈ ω ↦ if(𝑛 = ∅, (inr‘∅), (inl‘(𝑓 𝑛)))):ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
9186, 90syl 14 . . . 4 ((∃𝑥 𝑥𝐴𝑓:ω–onto𝐴) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
9291ex 115 . . 3 (∃𝑥 𝑥𝐴 → (𝑓:ω–onto𝐴 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o)))
9392exlimdv 1819 . 2 (∃𝑥 𝑥𝐴 → (∃𝑓 𝑓:ω–onto𝐴 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o)))
94 foeq1 5430 . . 3 (𝑓 = 𝑔 → (𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ 𝑔:ω–onto→(𝐴 ⊔ 1o)))
9594cbvexv 1918 . 2 (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
9693, 95syl6ibr 162 1 (∃𝑥 𝑥𝐴 → (∃𝑓 𝑓:ω–onto𝐴 → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834   = wceq 1353  wex 1492  wcel 2148  wral 2455  wrex 2456  c0 3422  ifcif 3534   cuni 3807  cmpt 4061  Tr wtr 4098  Ord word 4359  suc csuc 4362  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-fal 1359  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-ne 2348  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:  ctm  7102
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