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Theorem aceq3lem 8981
Description: Lemma for dfac3 8982. (Contributed by NM, 2-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypothesis
Ref Expression
aceq3lem.1 𝐹 = (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢𝑤𝑦𝑢}))
Assertion
Ref Expression
aceq3lem (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑢,𝑓
Allowed substitution hints:   𝐹(𝑥,𝑦,𝑧,𝑤,𝑢,𝑓)

Proof of Theorem aceq3lem
Dummy variables 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3234 . . . . . 6 𝑦 ∈ V
21rnex 7142 . . . . 5 ran 𝑦 ∈ V
32pwex 4878 . . . 4 𝒫 ran 𝑦 ∈ V
4 raleq 3168 . . . . 5 (𝑥 = 𝒫 ran 𝑦 → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
54exbidv 1890 . . . 4 (𝑥 = 𝒫 ran 𝑦 → (∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∃𝑓𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
63, 5spcv 3330 . . 3 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑓𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
7 aceq3lem.1 . . . . . . 7 𝐹 = (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢𝑤𝑦𝑢}))
8 df-mpt 4763 . . . . . . 7 (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢𝑤𝑦𝑢})) = {⟨𝑤, ⟩ ∣ (𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢}))}
97, 8eqtri 2673 . . . . . 6 𝐹 = {⟨𝑤, ⟩ ∣ (𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢}))}
10 vex 3234 . . . . . . . . . . . . . . 15 𝑤 ∈ V
1110eldm 5353 . . . . . . . . . . . . . 14 (𝑤 ∈ dom 𝑦 ↔ ∃𝑢 𝑤𝑦𝑢)
12 abn0 3987 . . . . . . . . . . . . . 14 ({𝑢𝑤𝑦𝑢} ≠ ∅ ↔ ∃𝑢 𝑤𝑦𝑢)
1311, 12bitr4i 267 . . . . . . . . . . . . 13 (𝑤 ∈ dom 𝑦 ↔ {𝑢𝑤𝑦𝑢} ≠ ∅)
14 vex 3234 . . . . . . . . . . . . . . . . 17 𝑢 ∈ V
1510, 14brelrn 5388 . . . . . . . . . . . . . . . 16 (𝑤𝑦𝑢𝑢 ∈ ran 𝑦)
1615abssi 3710 . . . . . . . . . . . . . . 15 {𝑢𝑤𝑦𝑢} ⊆ ran 𝑦
172elpw2 4858 . . . . . . . . . . . . . . 15 ({𝑢𝑤𝑦𝑢} ∈ 𝒫 ran 𝑦 ↔ {𝑢𝑤𝑦𝑢} ⊆ ran 𝑦)
1816, 17mpbir 221 . . . . . . . . . . . . . 14 {𝑢𝑤𝑦𝑢} ∈ 𝒫 ran 𝑦
19 neeq1 2885 . . . . . . . . . . . . . . . 16 (𝑧 = {𝑢𝑤𝑦𝑢} → (𝑧 ≠ ∅ ↔ {𝑢𝑤𝑦𝑢} ≠ ∅))
20 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑧 = {𝑢𝑤𝑦𝑢} → (𝑓𝑧) = (𝑓‘{𝑢𝑤𝑦𝑢}))
21 id 22 . . . . . . . . . . . . . . . . 17 (𝑧 = {𝑢𝑤𝑦𝑢} → 𝑧 = {𝑢𝑤𝑦𝑢})
2220, 21eleq12d 2724 . . . . . . . . . . . . . . . 16 (𝑧 = {𝑢𝑤𝑦𝑢} → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢}))
2319, 22imbi12d 333 . . . . . . . . . . . . . . 15 (𝑧 = {𝑢𝑤𝑦𝑢} → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ({𝑢𝑤𝑦𝑢} ≠ ∅ → (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢})))
2423rspcv 3336 . . . . . . . . . . . . . 14 ({𝑢𝑤𝑦𝑢} ∈ 𝒫 ran 𝑦 → (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ({𝑢𝑤𝑦𝑢} ≠ ∅ → (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢})))
2518, 24ax-mp 5 . . . . . . . . . . . . 13 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ({𝑢𝑤𝑦𝑢} ≠ ∅ → (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢}))
2613, 25syl5bi 232 . . . . . . . . . . . 12 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑤 ∈ dom 𝑦 → (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢}))
2726imp 444 . . . . . . . . . . 11 ((∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ 𝑤 ∈ dom 𝑦) → (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢})
28 fvex 6239 . . . . . . . . . . . 12 (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ V
29 breq2 4689 . . . . . . . . . . . 12 (𝑧 = (𝑓‘{𝑢𝑤𝑦𝑢}) → (𝑤𝑦𝑧𝑤𝑦(𝑓‘{𝑢𝑤𝑦𝑢})))
30 breq2 4689 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → (𝑤𝑦𝑢𝑤𝑦𝑧))
3130cbvabv 2776 . . . . . . . . . . . 12 {𝑢𝑤𝑦𝑢} = {𝑧𝑤𝑦𝑧}
3228, 29, 31elab2 3386 . . . . . . . . . . 11 ((𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢} ↔ 𝑤𝑦(𝑓‘{𝑢𝑤𝑦𝑢}))
3327, 32sylib 208 . . . . . . . . . 10 ((∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ 𝑤 ∈ dom 𝑦) → 𝑤𝑦(𝑓‘{𝑢𝑤𝑦𝑢}))
34 breq2 4689 . . . . . . . . . 10 ( = (𝑓‘{𝑢𝑤𝑦𝑢}) → (𝑤𝑦𝑤𝑦(𝑓‘{𝑢𝑤𝑦𝑢})))
3533, 34syl5ibrcom 237 . . . . . . . . 9 ((∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ 𝑤 ∈ dom 𝑦) → ( = (𝑓‘{𝑢𝑤𝑦𝑢}) → 𝑤𝑦))
3635expimpd 628 . . . . . . . 8 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ((𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢})) → 𝑤𝑦))
3736ssopab2dv 5033 . . . . . . 7 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → {⟨𝑤, ⟩ ∣ (𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢}))} ⊆ {⟨𝑤, ⟩ ∣ 𝑤𝑦})
38 opabss 4747 . . . . . . 7 {⟨𝑤, ⟩ ∣ 𝑤𝑦} ⊆ 𝑦
3937, 38syl6ss 3648 . . . . . 6 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → {⟨𝑤, ⟩ ∣ (𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢}))} ⊆ 𝑦)
409, 39syl5eqss 3682 . . . . 5 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → 𝐹𝑦)
4128, 7fnmpti 6060 . . . . 5 𝐹 Fn dom 𝑦
421ssex 4835 . . . . . . 7 (𝐹𝑦𝐹 ∈ V)
4342adantr 480 . . . . . 6 ((𝐹𝑦𝐹 Fn dom 𝑦) → 𝐹 ∈ V)
44 sseq1 3659 . . . . . . . 8 (𝑔 = 𝐹 → (𝑔𝑦𝐹𝑦))
45 fneq1 6017 . . . . . . . 8 (𝑔 = 𝐹 → (𝑔 Fn dom 𝑦𝐹 Fn dom 𝑦))
4644, 45anbi12d 747 . . . . . . 7 (𝑔 = 𝐹 → ((𝑔𝑦𝑔 Fn dom 𝑦) ↔ (𝐹𝑦𝐹 Fn dom 𝑦)))
4746spcegv 3325 . . . . . 6 (𝐹 ∈ V → ((𝐹𝑦𝐹 Fn dom 𝑦) → ∃𝑔(𝑔𝑦𝑔 Fn dom 𝑦)))
4843, 47mpcom 38 . . . . 5 ((𝐹𝑦𝐹 Fn dom 𝑦) → ∃𝑔(𝑔𝑦𝑔 Fn dom 𝑦))
4940, 41, 48sylancl 695 . . . 4 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑔(𝑔𝑦𝑔 Fn dom 𝑦))
5049exlimiv 1898 . . 3 (∃𝑓𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑔(𝑔𝑦𝑔 Fn dom 𝑦))
516, 50syl 17 . 2 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑔(𝑔𝑦𝑔 Fn dom 𝑦))
52 sseq1 3659 . . . 4 (𝑔 = 𝑓 → (𝑔𝑦𝑓𝑦))
53 fneq1 6017 . . . 4 (𝑔 = 𝑓 → (𝑔 Fn dom 𝑦𝑓 Fn dom 𝑦))
5452, 53anbi12d 747 . . 3 (𝑔 = 𝑓 → ((𝑔𝑦𝑔 Fn dom 𝑦) ↔ (𝑓𝑦𝑓 Fn dom 𝑦)))
5554cbvexv 2311 . 2 (∃𝑔(𝑔𝑦𝑔 Fn dom 𝑦) ↔ ∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦))
5651, 55sylib 208 1 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wne 2823  wral 2941  Vcvv 3231  wss 3607  c0 3948  𝒫 cpw 4191   class class class wbr 4685  {copab 4745  cmpt 4762  dom cdm 5143  ran crn 5144   Fn wfn 5921  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by:  dfac3  8982
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