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Theorem eliman0 6364
 Description: A non-nul 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 6356 . . . . 5 (𝐴𝐹(𝐹𝐴) ∨ (𝐹𝐴) = ∅)
2 orcom 850 . . . . 5 ((𝐴𝐹(𝐹𝐴) ∨ (𝐹𝐴) = ∅) ↔ ((𝐹𝐴) = ∅ ∨ 𝐴𝐹(𝐹𝐴)))
31, 2mpbi 220 . . . 4 ((𝐹𝐴) = ∅ ∨ 𝐴𝐹(𝐹𝐴))
43ori 841 . . 3 (¬ (𝐹𝐴) = ∅ → 𝐴𝐹(𝐹𝐴))
5 breq1 4787 . . . 4 (𝑥 = 𝐴 → (𝑥𝐹(𝐹𝐴) ↔ 𝐴𝐹(𝐹𝐴)))
65rspcev 3458 . . 3 ((𝐴𝐵𝐴𝐹(𝐹𝐴)) → ∃𝑥𝐵 𝑥𝐹(𝐹𝐴))
74, 6sylan2 572 . 2 ((𝐴𝐵 ∧ ¬ (𝐹𝐴) = ∅) → ∃𝑥𝐵 𝑥𝐹(𝐹𝐴))
8 fvex 6342 . . 3 (𝐹𝐴) ∈ V
98elima 5612 . 2 ((𝐹𝐴) ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 𝑥𝐹(𝐹𝐴))
107, 9sylibr 224 1 ((𝐴𝐵 ∧ ¬ (𝐹𝐴) = ∅) → (𝐹𝐴) ∈ (𝐹𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 382   ∨ wo 826   = wceq 1630   ∈ wcel 2144  ∃wrex 3061  ∅c0 4061   class class class wbr 4784   “ cima 5252  ‘cfv 6031 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fv 6039 This theorem is referenced by:  ovima0  6959  setrec2fun  42957
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