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Theorem aceq3lem 10118
Description: Lemma for dfac3 10119. (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 3477 . . . . . 6 𝑦 ∈ V
21rnex 7906 . . . . 5 ran 𝑦 ∈ V
32pwex 5378 . . . 4 𝒫 ran 𝑦 ∈ V
4 raleq 3321 . . . . 5 (𝑥 = 𝒫 ran 𝑦 → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
54exbidv 1923 . . . 4 (𝑥 = 𝒫 ran 𝑦 → (∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∃𝑓𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
63, 5spcv 3595 . . 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 5232 . . . . . . 7 (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢𝑤𝑦𝑢})) = {⟨𝑤, ⟩ ∣ (𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢}))}
97, 8eqtri 2759 . . . . . 6 𝐹 = {⟨𝑤, ⟩ ∣ (𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢}))}
10 vex 3477 . . . . . . . . . . . . . . 15 𝑤 ∈ V
1110eldm 5900 . . . . . . . . . . . . . 14 (𝑤 ∈ dom 𝑦 ↔ ∃𝑢 𝑤𝑦𝑢)
12 abn0 4380 . . . . . . . . . . . . . 14 ({𝑢𝑤𝑦𝑢} ≠ ∅ ↔ ∃𝑢 𝑤𝑦𝑢)
1311, 12bitr4i 278 . . . . . . . . . . . . 13 (𝑤 ∈ dom 𝑦 ↔ {𝑢𝑤𝑦𝑢} ≠ ∅)
14 vex 3477 . . . . . . . . . . . . . . . . 17 𝑢 ∈ V
1510, 14brelrn 5941 . . . . . . . . . . . . . . . 16 (𝑤𝑦𝑢𝑢 ∈ ran 𝑦)
1615abssi 4067 . . . . . . . . . . . . . . 15 {𝑢𝑤𝑦𝑢} ⊆ ran 𝑦
172, 16elpwi2 5346 . . . . . . . . . . . . . 14 {𝑢𝑤𝑦𝑢} ∈ 𝒫 ran 𝑦
18 neeq1 3002 . . . . . . . . . . . . . . . 16 (𝑧 = {𝑢𝑤𝑦𝑢} → (𝑧 ≠ ∅ ↔ {𝑢𝑤𝑦𝑢} ≠ ∅))
19 fveq2 6891 . . . . . . . . . . . . . . . . 17 (𝑧 = {𝑢𝑤𝑦𝑢} → (𝑓𝑧) = (𝑓‘{𝑢𝑤𝑦𝑢}))
20 id 22 . . . . . . . . . . . . . . . . 17 (𝑧 = {𝑢𝑤𝑦𝑢} → 𝑧 = {𝑢𝑤𝑦𝑢})
2119, 20eleq12d 2826 . . . . . . . . . . . . . . . 16 (𝑧 = {𝑢𝑤𝑦𝑢} → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢}))
2218, 21imbi12d 344 . . . . . . . . . . . . . . 15 (𝑧 = {𝑢𝑤𝑦𝑢} → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ({𝑢𝑤𝑦𝑢} ≠ ∅ → (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢})))
2322rspcv 3608 . . . . . . . . . . . . . 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, 24biimtrid 241 . . . . . . . . . . . 12 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑤 ∈ dom 𝑦 → (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢}))
2625imp 406 . . . . . . . . . . 11 ((∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ 𝑤 ∈ dom 𝑦) → (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢})
27 fvex 6904 . . . . . . . . . . . 12 (𝑓‘{𝑢𝑤𝑦𝑢}) ∈ V
28 breq2 5152 . . . . . . . . . . . 12 (𝑧 = (𝑓‘{𝑢𝑤𝑦𝑢}) → (𝑤𝑦𝑧𝑤𝑦(𝑓‘{𝑢𝑤𝑦𝑢})))
29 breq2 5152 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → (𝑤𝑦𝑢𝑤𝑦𝑧))
3029cbvabv 2804 . . . . . . . . . . . 12 {𝑢𝑤𝑦𝑢} = {𝑧𝑤𝑦𝑧}
3127, 28, 30elab2 3672 . . . . . . . . . . 11 ((𝑓‘{𝑢𝑤𝑦𝑢}) ∈ {𝑢𝑤𝑦𝑢} ↔ 𝑤𝑦(𝑓‘{𝑢𝑤𝑦𝑢}))
3226, 31sylib 217 . . . . . . . . . 10 ((∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ 𝑤 ∈ dom 𝑦) → 𝑤𝑦(𝑓‘{𝑢𝑤𝑦𝑢}))
33 breq2 5152 . . . . . . . . . 10 ( = (𝑓‘{𝑢𝑤𝑦𝑢}) → (𝑤𝑦𝑤𝑦(𝑓‘{𝑢𝑤𝑦𝑢})))
3432, 33syl5ibrcom 246 . . . . . . . . 9 ((∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ 𝑤 ∈ dom 𝑦) → ( = (𝑓‘{𝑢𝑤𝑦𝑢}) → 𝑤𝑦))
3534expimpd 453 . . . . . . . 8 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ((𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢})) → 𝑤𝑦))
3635ssopab2dv 5551 . . . . . . 7 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → {⟨𝑤, ⟩ ∣ (𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢}))} ⊆ {⟨𝑤, ⟩ ∣ 𝑤𝑦})
37 opabss 5212 . . . . . . 7 {⟨𝑤, ⟩ ∣ 𝑤𝑦} ⊆ 𝑦
3836, 37sstrdi 3994 . . . . . 6 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → {⟨𝑤, ⟩ ∣ (𝑤 ∈ dom 𝑦 = (𝑓‘{𝑢𝑤𝑦𝑢}))} ⊆ 𝑦)
399, 38eqsstrid 4030 . . . . 5 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → 𝐹𝑦)
4027, 7fnmpti 6693 . . . . 5 𝐹 Fn dom 𝑦
411ssex 5321 . . . . . . 7 (𝐹𝑦𝐹 ∈ V)
4241adantr 480 . . . . . 6 ((𝐹𝑦𝐹 Fn dom 𝑦) → 𝐹 ∈ V)
43 sseq1 4007 . . . . . . . 8 (𝑔 = 𝐹 → (𝑔𝑦𝐹𝑦))
44 fneq1 6640 . . . . . . . 8 (𝑔 = 𝐹 → (𝑔 Fn dom 𝑦𝐹 Fn dom 𝑦))
4543, 44anbi12d 630 . . . . . . 7 (𝑔 = 𝐹 → ((𝑔𝑦𝑔 Fn dom 𝑦) ↔ (𝐹𝑦𝐹 Fn dom 𝑦)))
4645spcegv 3587 . . . . . 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 585 . . . 4 (∀𝑧 ∈ 𝒫 ran 𝑦(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑔(𝑔𝑦𝑔 Fn dom 𝑦))
4948exlimiv 1932 . . 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 4007 . . . 4 (𝑔 = 𝑓 → (𝑔𝑦𝑓𝑦))
52 fneq1 6640 . . . 4 (𝑔 = 𝑓 → (𝑔 Fn dom 𝑦𝑓 Fn dom 𝑦))
5351, 52anbi12d 630 . . 3 (𝑔 = 𝑓 → ((𝑔𝑦𝑔 Fn dom 𝑦) ↔ (𝑓𝑦𝑓 Fn dom 𝑦)))
5453cbvexvw 2039 . 2 (∃𝑔(𝑔𝑦𝑔 Fn dom 𝑦) ↔ ∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦))
5550, 54sylib 217 1 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1538   = wceq 1540  wex 1780  wcel 2105  {cab 2708  wne 2939  wral 3060  Vcvv 3473  wss 3948  c0 4322  𝒫 cpw 4602   class class class wbr 5148  {copab 5210  cmpt 5231  dom cdm 5676  ran crn 5677   Fn wfn 6538  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551
This theorem is referenced by:  dfac3  10119
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