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Theorem aceq3lem 9540
Description: Lemma for dfac3 9541. (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 3497 . . . . . 6 𝑦 ∈ V
21rnex 7611 . . . . 5 ran 𝑦 ∈ V
32pwex 5273 . . . 4 𝒫 ran 𝑦 ∈ V
4 raleq 3405 . . . . 5 (𝑥 = 𝒫 ran 𝑦 → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
54exbidv 1918 . . . 4 (𝑥 = 𝒫 ran 𝑦 → (∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∃𝑓𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
63, 5spcv 3605 . . 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 5139 . . . . . . 7 (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢𝑤𝑦𝑢})) = {⟨𝑤, ⟩ ∣ (𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢}))}
97, 8eqtri 2844 . . . . . 6 𝐹 = {⟨𝑤, ⟩ ∣ (𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢}))}
10 vex 3497 . . . . . . . . . . . . . . 15 𝑤 ∈ V
1110eldm 5763 . . . . . . . . . . . . . 14 (𝑤 ∈ dom 𝑦 ↔ ∃𝑢 𝑤𝑦𝑢)
12 abn0 4335 . . . . . . . . . . . . . 14 ({𝑢𝑤𝑦𝑢} ≠ ∅ ↔ ∃𝑢 𝑤𝑦𝑢)
1311, 12bitr4i 280 . . . . . . . . . . . . 13 (𝑤 ∈ dom 𝑦 ↔ {𝑢𝑤𝑦𝑢} ≠ ∅)
14 vex 3497 . . . . . . . . . . . . . . . . 17 𝑢 ∈ V
1510, 14brelrn 5806 . . . . . . . . . . . . . . . 16 (𝑤𝑦𝑢𝑢 ∈ ran 𝑦)
1615abssi 4045 . . . . . . . . . . . . . . 15 {𝑢𝑤𝑦𝑢} ⊆ ran 𝑦
172, 16elpwi2 5241 . . . . . . . . . . . . . 14 {𝑢𝑤𝑦𝑢} ∈ 𝒫 ran 𝑦
18 neeq1 3078 . . . . . . . . . . . . . . . 16 (𝑧 = {𝑢𝑤𝑦𝑢} → (𝑧 ≠ ∅ ↔ {𝑢𝑤𝑦𝑢} ≠ ∅))
19 fveq2 6664 . . . . . . . . . . . . . . . . 17 (𝑧 = {𝑢𝑤𝑦𝑢} → (𝑓𝑧) = (𝑓‘{𝑢𝑤𝑦𝑢}))
20 id 22 . . . . . . . . . . . . . . . . 17 (𝑧 = {𝑢𝑤𝑦𝑢} → 𝑧 = {𝑢𝑤𝑦𝑢})
2119, 20eleq12d 2907 . . . . . . . . . . . . . . . 16 (𝑧 = {𝑢𝑤𝑦𝑢} → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢}))
2218, 21imbi12d 347 . . . . . . . . . . . . . . 15 (𝑧 = {𝑢𝑤𝑦𝑢} → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ({𝑢𝑤𝑦𝑢} ≠ ∅ → (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢})))
2322rspcv 3617 . . . . . . . . . . . . . 14 ({𝑢𝑤𝑦𝑢} ∈ 𝒫 ran 𝑦 → (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ({𝑢𝑤𝑦𝑢} ≠ ∅ → (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢})))
2417, 23ax-mp 5 . . . . . . . . . . . . 13 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ({𝑢𝑤𝑦𝑢} ≠ ∅ → (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢}))
2513, 24syl5bi 244 . . . . . . . . . . . 12 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑤 ∈ dom 𝑦 → (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢}))
2625imp 409 . . . . . . . . . . 11 ((∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ 𝑤 ∈ dom 𝑦) → (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢})
27 fvex 6677 . . . . . . . . . . . 12 (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ V
28 breq2 5062 . . . . . . . . . . . 12 (𝑧 = (𝑓‘{𝑢𝑤𝑦𝑢}) → (𝑤𝑦𝑧𝑤𝑦(𝑓‘{𝑢𝑤𝑦𝑢})))
29 breq2 5062 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → (𝑤𝑦𝑢𝑤𝑦𝑧))
3029cbvabv 2889 . . . . . . . . . . . 12 {𝑢𝑤𝑦𝑢} = {𝑧𝑤𝑦𝑧}
3127, 28, 30elab2 3669 . . . . . . . . . . 11 ((𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢} ↔ 𝑤𝑦(𝑓‘{𝑢𝑤𝑦𝑢}))
3226, 31sylib 220 . . . . . . . . . 10 ((∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ 𝑤 ∈ dom 𝑦) → 𝑤𝑦(𝑓‘{𝑢𝑤𝑦𝑢}))
33 breq2 5062 . . . . . . . . . 10 ( = (𝑓‘{𝑢𝑤𝑦𝑢}) → (𝑤𝑦𝑤𝑦(𝑓‘{𝑢𝑤𝑦𝑢})))
3432, 33syl5ibrcom 249 . . . . . . . . 9 ((∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ 𝑤 ∈ dom 𝑦) → ( = (𝑓‘{𝑢𝑤𝑦𝑢}) → 𝑤𝑦))
3534expimpd 456 . . . . . . . 8 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ((𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢})) → 𝑤𝑦))
3635ssopab2dv 5430 . . . . . . 7 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → {⟨𝑤, ⟩ ∣ (𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢}))} ⊆ {⟨𝑤, ⟩ ∣ 𝑤𝑦})
37 opabss 5122 . . . . . . 7 {⟨𝑤, ⟩ ∣ 𝑤𝑦} ⊆ 𝑦
3836, 37sstrdi 3978 . . . . . 6 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → {⟨𝑤, ⟩ ∣ (𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢}))} ⊆ 𝑦)
399, 38eqsstrid 4014 . . . . 5 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → 𝐹𝑦)
4027, 7fnmpti 6485 . . . . 5 𝐹 Fn dom 𝑦
411ssex 5217 . . . . . . 7 (𝐹𝑦𝐹 ∈ V)
4241adantr 483 . . . . . 6 ((𝐹𝑦𝐹 Fn dom 𝑦) → 𝐹 ∈ V)
43 sseq1 3991 . . . . . . . 8 (𝑔 = 𝐹 → (𝑔𝑦𝐹𝑦))
44 fneq1 6438 . . . . . . . 8 (𝑔 = 𝐹 → (𝑔 Fn dom 𝑦𝐹 Fn dom 𝑦))
4543, 44anbi12d 632 . . . . . . 7 (𝑔 = 𝐹 → ((𝑔𝑦𝑔 Fn dom 𝑦) ↔ (𝐹𝑦𝐹 Fn dom 𝑦)))
4645spcegv 3596 . . . . . 6 (𝐹 ∈ V → ((𝐹𝑦𝐹 Fn dom 𝑦) → ∃𝑔(𝑔𝑦𝑔 Fn dom 𝑦)))
4742, 46mpcom 38 . . . . 5 ((𝐹𝑦𝐹 Fn dom 𝑦) → ∃𝑔(𝑔𝑦𝑔 Fn dom 𝑦))
4839, 40, 47sylancl 588 . . . 4 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑔(𝑔𝑦𝑔 Fn dom 𝑦))
4948exlimiv 1927 . . 3 (∃𝑓𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑔(𝑔𝑦𝑔 Fn dom 𝑦))
506, 49syl 17 . 2 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑔(𝑔𝑦𝑔 Fn dom 𝑦))
51 sseq1 3991 . . . 4 (𝑔 = 𝑓 → (𝑔𝑦𝑓𝑦))
52 fneq1 6438 . . . 4 (𝑔 = 𝑓 → (𝑔 Fn dom 𝑦𝑓 Fn dom 𝑦))
5351, 52anbi12d 632 . . 3 (𝑔 = 𝑓 → ((𝑔𝑦𝑔 Fn dom 𝑦) ↔ (𝑓𝑦𝑓 Fn dom 𝑦)))
5453cbvexvw 2040 . 2 (∃𝑔(𝑔𝑦𝑔 Fn dom 𝑦) ↔ ∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦))
5550, 54sylib 220 1 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wal 1531   = wceq 1533  wex 1776  wcel 2110  {cab 2799  wne 3016  wral 3138  Vcvv 3494  wss 3935  c0 4290  𝒫 cpw 4538   class class class wbr 5058  {copab 5120  cmpt 5138  dom cdm 5549  ran crn 5550   Fn wfn 6344  cfv 6349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-iota 6308  df-fun 6351  df-fn 6352  df-fv 6357
This theorem is referenced by:  dfac3  9541
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