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Theorem bj-fvimacnv0 34560
Description: Variant of fvimacnv 6816 where membership of 𝐴 in the domain is not needed provided the containing class 𝐵 does not contain the empty set. Note that this antecedent would not be needed with definition df-afv 43309. (Contributed by BJ, 7-Jan-2024.)
Assertion
Ref Expression
bj-fvimacnv0 ((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))

Proof of Theorem bj-fvimacnv0
StepHypRef Expression
1 eleq1 2898 . . . . . . . . . 10 ((𝐹𝐴) = ∅ → ((𝐹𝐴) ∈ 𝐵 ↔ ∅ ∈ 𝐵))
21biimpcd 251 . . . . . . . . 9 ((𝐹𝐴) ∈ 𝐵 → ((𝐹𝐴) = ∅ → ∅ ∈ 𝐵))
32con3rr3 158 . . . . . . . 8 (¬ ∅ ∈ 𝐵 → ((𝐹𝐴) ∈ 𝐵 → ¬ (𝐹𝐴) = ∅))
43imp 409 . . . . . . 7 ((¬ ∅ ∈ 𝐵 ∧ (𝐹𝐴) ∈ 𝐵) → ¬ (𝐹𝐴) = ∅)
5 ndmfv 6693 . . . . . . 7 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
64, 5nsyl2 143 . . . . . 6 ((¬ ∅ ∈ 𝐵 ∧ (𝐹𝐴) ∈ 𝐵) → 𝐴 ∈ dom 𝐹)
7 simpr 487 . . . . . 6 ((¬ ∅ ∈ 𝐵 ∧ (𝐹𝐴) ∈ 𝐵) → (𝐹𝐴) ∈ 𝐵)
8 fvimacnv 6816 . . . . . . . . 9 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))
98biimpd 231 . . . . . . . 8 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))
109ex 415 . . . . . . 7 (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵))))
1110com3l 89 . . . . . 6 (𝐴 ∈ dom 𝐹 → ((𝐹𝐴) ∈ 𝐵 → (Fun 𝐹𝐴 ∈ (𝐹𝐵))))
126, 7, 11sylc 65 . . . . 5 ((¬ ∅ ∈ 𝐵 ∧ (𝐹𝐴) ∈ 𝐵) → (Fun 𝐹𝐴 ∈ (𝐹𝐵)))
1312ex 415 . . . 4 (¬ ∅ ∈ 𝐵 → ((𝐹𝐴) ∈ 𝐵 → (Fun 𝐹𝐴 ∈ (𝐹𝐵))))
1413com3r 87 . . 3 (Fun 𝐹 → (¬ ∅ ∈ 𝐵 → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵))))
1514imp 409 . 2 ((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))
16 fvimacnvi 6815 . . . 4 ((Fun 𝐹𝐴 ∈ (𝐹𝐵)) → (𝐹𝐴) ∈ 𝐵)
1716ex 415 . . 3 (Fun 𝐹 → (𝐴 ∈ (𝐹𝐵) → (𝐹𝐴) ∈ 𝐵))
1817adantr 483 . 2 ((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → (𝐴 ∈ (𝐹𝐵) → (𝐹𝐴) ∈ 𝐵))
1915, 18impbid 214 1 ((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1530  wcel 2107  c0 4289  ccnv 5547  dom cdm 5548  cima 5551  Fun wfun 6342  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356
This theorem is referenced by:  bj-isrvec  34567
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