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Theorem funimaexgOLD 6528
Description: Obsolete version of funimaexg 6527 as of 19-Dec-2024. (Contributed by NM, 10-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
funimaexgOLD ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)

Proof of Theorem funimaexgOLD
Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imaeq2 5968 . . . . 5 (𝑤 = 𝐵 → (𝐴𝑤) = (𝐴𝐵))
21eleq1d 2824 . . . 4 (𝑤 = 𝐵 → ((𝐴𝑤) ∈ V ↔ (𝐴𝐵) ∈ V))
32imbi2d 341 . . 3 (𝑤 = 𝐵 → ((Fun 𝐴 → (𝐴𝑤) ∈ V) ↔ (Fun 𝐴 → (𝐴𝐵) ∈ V)))
4 dffun5 6453 . . . 4 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧)))
5 nfv 1918 . . . . . 6 𝑧𝑥, 𝑦⟩ ∈ 𝐴
65axrep4 5215 . . . . 5 (∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
7 isset 3446 . . . . . 6 ((𝐴𝑤) ∈ V ↔ ∃𝑧 𝑧 = (𝐴𝑤))
8 dfima3 5975 . . . . . . . . 9 (𝐴𝑤) = {𝑦 ∣ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
98eqeq2i 2752 . . . . . . . 8 (𝑧 = (𝐴𝑤) ↔ 𝑧 = {𝑦 ∣ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)})
10 abeq2 2873 . . . . . . . 8 (𝑧 = {𝑦 ∣ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} ↔ ∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
119, 10bitri 274 . . . . . . 7 (𝑧 = (𝐴𝑤) ↔ ∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
1211exbii 1851 . . . . . 6 (∃𝑧 𝑧 = (𝐴𝑤) ↔ ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
137, 12bitri 274 . . . . 5 ((𝐴𝑤) ∈ V ↔ ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
146, 13sylibr 233 . . . 4 (∀𝑥𝑧𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴𝑦 = 𝑧) → (𝐴𝑤) ∈ V)
154, 14simplbiim 505 . . 3 (Fun 𝐴 → (𝐴𝑤) ∈ V)
163, 15vtoclg 3506 . 2 (𝐵𝐶 → (Fun 𝐴 → (𝐴𝐵) ∈ V))
1716impcom 408 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1537   = wceq 1539  wex 1782  wcel 2107  {cab 2716  Vcvv 3433  cop 4568  cima 5593  Rel wrel 5595  Fun wfun 6431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-rep 5210  ax-sep 5224  ax-nul 5231  ax-pr 5353
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3435  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-br 5076  df-opab 5138  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-fun 6439
This theorem is referenced by: (None)
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