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Theorem eliman0 6696
Description: A nonempty function value is an element of the image of the function. (Contributed by Thierry Arnoux, 25-Jun-2019.)
Assertion
Ref Expression
eliman0 ((𝐴𝐵 ∧ ¬ (𝐹𝐴) = ∅) → (𝐹𝐴) ∈ (𝐹𝐵))

Proof of Theorem eliman0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fvbr0 6688 . . . . 5 (𝐴𝐹(𝐹𝐴) ∨ (𝐹𝐴) = ∅)
2 orcom 867 . . . . 5 ((𝐴𝐹(𝐹𝐴) ∨ (𝐹𝐴) = ∅) ↔ ((𝐹𝐴) = ∅ ∨ 𝐴𝐹(𝐹𝐴)))
31, 2mpbi 233 . . . 4 ((𝐹𝐴) = ∅ ∨ 𝐴𝐹(𝐹𝐴))
43ori 858 . . 3 (¬ (𝐹𝐴) = ∅ → 𝐴𝐹(𝐹𝐴))
5 breq1 5055 . . . 4 (𝑥 = 𝐴 → (𝑥𝐹(𝐹𝐴) ↔ 𝐴𝐹(𝐹𝐴)))
65rspcev 3609 . . 3 ((𝐴𝐵𝐴𝐹(𝐹𝐴)) → ∃𝑥𝐵 𝑥𝐹(𝐹𝐴))
74, 6sylan2 595 . 2 ((𝐴𝐵 ∧ ¬ (𝐹𝐴) = ∅) → ∃𝑥𝐵 𝑥𝐹(𝐹𝐴))
8 fvex 6674 . . 3 (𝐹𝐴) ∈ V
98elima 5921 . 2 ((𝐹𝐴) ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 𝑥𝐹(𝐹𝐴))
107, 9sylibr 237 1 ((𝐴𝐵 ∧ ¬ (𝐹𝐴) = ∅) → (𝐹𝐴) ∈ (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 844   = wceq 1538  wcel 2115  wrex 3134  c0 4276   class class class wbr 5052  cima 5545  cfv 6343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-xp 5548  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fv 6351
This theorem is referenced by:  ovima0  7321  setrec2fun  45152
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