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Theorem aceq3lem 9143
Description: Lemma for dfac3 9144. (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 3354 . . . . . 6 𝑦 ∈ V
21rnex 7247 . . . . 5 ran 𝑦 ∈ V
32pwex 4981 . . . 4 𝒫 ran 𝑦 ∈ V
4 raleq 3287 . . . . 5 (𝑥 = 𝒫 ran 𝑦 → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
54exbidv 2002 . . . 4 (𝑥 = 𝒫 ran 𝑦 → (∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∃𝑓𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
63, 5spcv 3450 . . 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 4864 . . . . . . 7 (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢𝑤𝑦𝑢})) = {⟨𝑤, ⟩ ∣ (𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢}))}
97, 8eqtri 2793 . . . . . 6 𝐹 = {⟨𝑤, ⟩ ∣ (𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢}))}
10 vex 3354 . . . . . . . . . . . . . . 15 𝑤 ∈ V
1110eldm 5459 . . . . . . . . . . . . . 14 (𝑤 ∈ dom 𝑦 ↔ ∃𝑢 𝑤𝑦𝑢)
12 abn0 4101 . . . . . . . . . . . . . 14 ({𝑢𝑤𝑦𝑢} ≠ ∅ ↔ ∃𝑢 𝑤𝑦𝑢)
1311, 12bitr4i 267 . . . . . . . . . . . . 13 (𝑤 ∈ dom 𝑦 ↔ {𝑢𝑤𝑦𝑢} ≠ ∅)
14 vex 3354 . . . . . . . . . . . . . . . . 17 𝑢 ∈ V
1510, 14brelrn 5494 . . . . . . . . . . . . . . . 16 (𝑤𝑦𝑢𝑢 ∈ ran 𝑦)
1615abssi 3826 . . . . . . . . . . . . . . 15 {𝑢𝑤𝑦𝑢} ⊆ ran 𝑦
172elpw2 4959 . . . . . . . . . . . . . . 15 ({𝑢𝑤𝑦𝑢} ∈ 𝒫 ran 𝑦 ↔ {𝑢𝑤𝑦𝑢} ⊆ ran 𝑦)
1816, 17mpbir 221 . . . . . . . . . . . . . 14 {𝑢𝑤𝑦𝑢} ∈ 𝒫 ran 𝑦
19 neeq1 3005 . . . . . . . . . . . . . . . 16 (𝑧 = {𝑢𝑤𝑦𝑢} → (𝑧 ≠ ∅ ↔ {𝑢𝑤𝑦𝑢} ≠ ∅))
20 fveq2 6332 . . . . . . . . . . . . . . . . 17 (𝑧 = {𝑢𝑤𝑦𝑢} → (𝑓𝑧) = (𝑓‘{𝑢𝑤𝑦𝑢}))
21 id 22 . . . . . . . . . . . . . . . . 17 (𝑧 = {𝑢𝑤𝑦𝑢} → 𝑧 = {𝑢𝑤𝑦𝑢})
2220, 21eleq12d 2844 . . . . . . . . . . . . . . . 16 (𝑧 = {𝑢𝑤𝑦𝑢} → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢}))
2319, 22imbi12d 333 . . . . . . . . . . . . . . 15 (𝑧 = {𝑢𝑤𝑦𝑢} → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ({𝑢𝑤𝑦𝑢} ≠ ∅ → (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢})))
2423rspcv 3456 . . . . . . . . . . . . . 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 393 . . . . . . . . . . 11 ((∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ 𝑤 ∈ dom 𝑦) → (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢})
28 fvex 6342 . . . . . . . . . . . 12 (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ V
29 breq2 4790 . . . . . . . . . . . 12 (𝑧 = (𝑓‘{𝑢𝑤𝑦𝑢}) → (𝑤𝑦𝑧𝑤𝑦(𝑓‘{𝑢𝑤𝑦𝑢})))
30 breq2 4790 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → (𝑤𝑦𝑢𝑤𝑦𝑧))
3130cbvabv 2896 . . . . . . . . . . . 12 {𝑢𝑤𝑦𝑢} = {𝑧𝑤𝑦𝑧}
3228, 29, 31elab2 3505 . . . . . . . . . . 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 4790 . . . . . . . . . 10 ( = (𝑓‘{𝑢𝑤𝑦𝑢}) → (𝑤𝑦𝑤𝑦(𝑓‘{𝑢𝑤𝑦𝑢})))
3533, 34syl5ibrcom 237 . . . . . . . . 9 ((∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ 𝑤 ∈ dom 𝑦) → ( = (𝑓‘{𝑢𝑤𝑦𝑢}) → 𝑤𝑦))
3635expimpd 441 . . . . . . . 8 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ((𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢})) → 𝑤𝑦))
3736ssopab2dv 5137 . . . . . . 7 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → {⟨𝑤, ⟩ ∣ (𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢}))} ⊆ {⟨𝑤, ⟩ ∣ 𝑤𝑦})
38 opabss 4848 . . . . . . 7 {⟨𝑤, ⟩ ∣ 𝑤𝑦} ⊆ 𝑦
3937, 38syl6ss 3764 . . . . . 6 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → {⟨𝑤, ⟩ ∣ (𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢}))} ⊆ 𝑦)
409, 39syl5eqss 3798 . . . . 5 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → 𝐹𝑦)
4128, 7fnmpti 6162 . . . . 5 𝐹 Fn dom 𝑦
421ssex 4936 . . . . . . 7 (𝐹𝑦𝐹 ∈ V)
4342adantr 466 . . . . . 6 ((𝐹𝑦𝐹 Fn dom 𝑦) → 𝐹 ∈ V)
44 sseq1 3775 . . . . . . . 8 (𝑔 = 𝐹 → (𝑔𝑦𝐹𝑦))
45 fneq1 6119 . . . . . . . 8 (𝑔 = 𝐹 → (𝑔 Fn dom 𝑦𝐹 Fn dom 𝑦))
4644, 45anbi12d 616 . . . . . . 7 (𝑔 = 𝐹 → ((𝑔𝑦𝑔 Fn dom 𝑦) ↔ (𝐹𝑦𝐹 Fn dom 𝑦)))
4746spcegv 3445 . . . . . 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 574 . . . 4 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑔(𝑔𝑦𝑔 Fn dom 𝑦))
5049exlimiv 2010 . . 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 3775 . . . 4 (𝑔 = 𝑓 → (𝑔𝑦𝑓𝑦))
53 fneq1 6119 . . . 4 (𝑔 = 𝑓 → (𝑔 Fn dom 𝑦𝑓 Fn dom 𝑦))
5452, 53anbi12d 616 . . 3 (𝑔 = 𝑓 → ((𝑔𝑦𝑔 Fn dom 𝑦) ↔ (𝑓𝑦𝑓 Fn dom 𝑦)))
5554cbvexvw 2126 . 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 382  wal 1629   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wne 2943  wral 3061  Vcvv 3351  wss 3723  c0 4063  𝒫 cpw 4297   class class class wbr 4786  {copab 4846  cmpt 4863  dom cdm 5249  ran crn 5250   Fn wfn 6026  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039
This theorem is referenced by:  dfac3  9144
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