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Theorem aceq3lem 9226
Description: Lemma for dfac3 9227. (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 3394 . . . . . 6 𝑦 ∈ V
21rnex 7330 . . . . 5 ran 𝑦 ∈ V
32pwex 5050 . . . 4 𝒫 ran 𝑦 ∈ V
4 raleq 3327 . . . . 5 (𝑥 = 𝒫 ran 𝑦 → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
54exbidv 2012 . . . 4 (𝑥 = 𝒫 ran 𝑦 → (∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∃𝑓𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
63, 5spcv 3492 . . 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 4924 . . . . . . 7 (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢𝑤𝑦𝑢})) = {⟨𝑤, ⟩ ∣ (𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢}))}
97, 8eqtri 2828 . . . . . 6 𝐹 = {⟨𝑤, ⟩ ∣ (𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢}))}
10 vex 3394 . . . . . . . . . . . . . . 15 𝑤 ∈ V
1110eldm 5522 . . . . . . . . . . . . . 14 (𝑤 ∈ dom 𝑦 ↔ ∃𝑢 𝑤𝑦𝑢)
12 abn0 4155 . . . . . . . . . . . . . 14 ({𝑢𝑤𝑦𝑢} ≠ ∅ ↔ ∃𝑢 𝑤𝑦𝑢)
1311, 12bitr4i 269 . . . . . . . . . . . . 13 (𝑤 ∈ dom 𝑦 ↔ {𝑢𝑤𝑦𝑢} ≠ ∅)
14 vex 3394 . . . . . . . . . . . . . . . . 17 𝑢 ∈ V
1510, 14brelrn 5557 . . . . . . . . . . . . . . . 16 (𝑤𝑦𝑢𝑢 ∈ ran 𝑦)
1615abssi 3874 . . . . . . . . . . . . . . 15 {𝑢𝑤𝑦𝑢} ⊆ ran 𝑦
172elpw2 5020 . . . . . . . . . . . . . . 15 ({𝑢𝑤𝑦𝑢} ∈ 𝒫 ran 𝑦 ↔ {𝑢𝑤𝑦𝑢} ⊆ ran 𝑦)
1816, 17mpbir 222 . . . . . . . . . . . . . 14 {𝑢𝑤𝑦𝑢} ∈ 𝒫 ran 𝑦
19 neeq1 3040 . . . . . . . . . . . . . . . 16 (𝑧 = {𝑢𝑤𝑦𝑢} → (𝑧 ≠ ∅ ↔ {𝑢𝑤𝑦𝑢} ≠ ∅))
20 fveq2 6408 . . . . . . . . . . . . . . . . 17 (𝑧 = {𝑢𝑤𝑦𝑢} → (𝑓𝑧) = (𝑓‘{𝑢𝑤𝑦𝑢}))
21 id 22 . . . . . . . . . . . . . . . . 17 (𝑧 = {𝑢𝑤𝑦𝑢} → 𝑧 = {𝑢𝑤𝑦𝑢})
2220, 21eleq12d 2879 . . . . . . . . . . . . . . . 16 (𝑧 = {𝑢𝑤𝑦𝑢} → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢}))
2319, 22imbi12d 335 . . . . . . . . . . . . . . 15 (𝑧 = {𝑢𝑤𝑦𝑢} → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ({𝑢𝑤𝑦𝑢} ≠ ∅ → (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢})))
2423rspcv 3498 . . . . . . . . . . . . . 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 233 . . . . . . . . . . . 12 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑤 ∈ dom 𝑦 → (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢}))
2726imp 395 . . . . . . . . . . 11 ((∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ 𝑤 ∈ dom 𝑦) → (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢})
28 fvex 6421 . . . . . . . . . . . 12 (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ V
29 breq2 4848 . . . . . . . . . . . 12 (𝑧 = (𝑓‘{𝑢𝑤𝑦𝑢}) → (𝑤𝑦𝑧𝑤𝑦(𝑓‘{𝑢𝑤𝑦𝑢})))
30 breq2 4848 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → (𝑤𝑦𝑢𝑤𝑦𝑧))
3130cbvabv 2931 . . . . . . . . . . . 12 {𝑢𝑤𝑦𝑢} = {𝑧𝑤𝑦𝑧}
3228, 29, 31elab2 3549 . . . . . . . . . . 11 ((𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢} ↔ 𝑤𝑦(𝑓‘{𝑢𝑤𝑦𝑢}))
3327, 32sylib 209 . . . . . . . . . 10 ((∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ 𝑤 ∈ dom 𝑦) → 𝑤𝑦(𝑓‘{𝑢𝑤𝑦𝑢}))
34 breq2 4848 . . . . . . . . . 10 ( = (𝑓‘{𝑢𝑤𝑦𝑢}) → (𝑤𝑦𝑤𝑦(𝑓‘{𝑢𝑤𝑦𝑢})))
3533, 34syl5ibrcom 238 . . . . . . . . 9 ((∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ 𝑤 ∈ dom 𝑦) → ( = (𝑓‘{𝑢𝑤𝑦𝑢}) → 𝑤𝑦))
3635expimpd 443 . . . . . . . 8 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ((𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢})) → 𝑤𝑦))
3736ssopab2dv 5199 . . . . . . 7 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → {⟨𝑤, ⟩ ∣ (𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢}))} ⊆ {⟨𝑤, ⟩ ∣ 𝑤𝑦})
38 opabss 4908 . . . . . . 7 {⟨𝑤, ⟩ ∣ 𝑤𝑦} ⊆ 𝑦
3937, 38syl6ss 3810 . . . . . 6 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → {⟨𝑤, ⟩ ∣ (𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢}))} ⊆ 𝑦)
409, 39syl5eqss 3846 . . . . 5 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → 𝐹𝑦)
4128, 7fnmpti 6233 . . . . 5 𝐹 Fn dom 𝑦
421ssex 4997 . . . . . . 7 (𝐹𝑦𝐹 ∈ V)
4342adantr 468 . . . . . 6 ((𝐹𝑦𝐹 Fn dom 𝑦) → 𝐹 ∈ V)
44 sseq1 3823 . . . . . . . 8 (𝑔 = 𝐹 → (𝑔𝑦𝐹𝑦))
45 fneq1 6190 . . . . . . . 8 (𝑔 = 𝐹 → (𝑔 Fn dom 𝑦𝐹 Fn dom 𝑦))
4644, 45anbi12d 618 . . . . . . 7 (𝑔 = 𝐹 → ((𝑔𝑦𝑔 Fn dom 𝑦) ↔ (𝐹𝑦𝐹 Fn dom 𝑦)))
4746spcegv 3487 . . . . . 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 576 . . . 4 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑔(𝑔𝑦𝑔 Fn dom 𝑦))
5049exlimiv 2021 . . 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 3823 . . . 4 (𝑔 = 𝑓 → (𝑔𝑦𝑓𝑦))
53 fneq1 6190 . . . 4 (𝑔 = 𝑓 → (𝑔 Fn dom 𝑦𝑓 Fn dom 𝑦))
5452, 53anbi12d 618 . . 3 (𝑔 = 𝑓 → ((𝑔𝑦𝑔 Fn dom 𝑦) ↔ (𝑓𝑦𝑓 Fn dom 𝑦)))
5554cbvexvw 2137 . 2 (∃𝑔(𝑔𝑦𝑔 Fn dom 𝑦) ↔ ∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦))
5651, 55sylib 209 1 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1635   = wceq 1637  wex 1859  wcel 2156  {cab 2792  wne 2978  wral 3096  Vcvv 3391  wss 3769  c0 4116  𝒫 cpw 4351   class class class wbr 4844  {copab 4906  cmpt 4923  dom cdm 5311  ran crn 5312   Fn wfn 6096  cfv 6101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-iota 6064  df-fun 6103  df-fn 6104  df-fv 6109
This theorem is referenced by:  dfac3  9227
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