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Theorem preimafvn0 46782
Description: The preimage of a function value is not empty. (Contributed by AV, 7-Mar-2024.)
Assertion
Ref Expression
preimafvn0 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹 “ {(𝐹𝑋)}) ≠ ∅)

Proof of Theorem preimafvn0
StepHypRef Expression
1 preimafvsnel 46781 . 2 ((𝐹 Fn 𝐴𝑋𝐴) → 𝑋 ∈ (𝐹 “ {(𝐹𝑋)}))
21ne0d 4331 1 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹 “ {(𝐹𝑋)}) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098  wne 2930  c0 4318  {csn 4624  ccnv 5671  cima 5675   Fn wfn 6537  cfv 6542
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550
This theorem is referenced by: (None)
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