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Theorem eliman0 6133
 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 6125 . . . . 5 (𝐴𝐹(𝐹𝐴) ∨ (𝐹𝐴) = ∅)
2 orcom 401 . . . . 5 ((𝐴𝐹(𝐹𝐴) ∨ (𝐹𝐴) = ∅) ↔ ((𝐹𝐴) = ∅ ∨ 𝐴𝐹(𝐹𝐴)))
31, 2mpbi 219 . . . 4 ((𝐹𝐴) = ∅ ∨ 𝐴𝐹(𝐹𝐴))
43ori 389 . . 3 (¬ (𝐹𝐴) = ∅ → 𝐴𝐹(𝐹𝐴))
5 breq1 4586 . . . 4 (𝑥 = 𝐴 → (𝑥𝐹(𝐹𝐴) ↔ 𝐴𝐹(𝐹𝐴)))
65rspcev 3282 . . 3 ((𝐴𝐵𝐴𝐹(𝐹𝐴)) → ∃𝑥𝐵 𝑥𝐹(𝐹𝐴))
74, 6sylan2 490 . 2 ((𝐴𝐵 ∧ ¬ (𝐹𝐴) = ∅) → ∃𝑥𝐵 𝑥𝐹(𝐹𝐴))
8 fvex 6113 . . 3 (𝐹𝐴) ∈ V
98elima 5390 . 2 ((𝐹𝐴) ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 𝑥𝐹(𝐹𝐴))
107, 9sylibr 223 1 ((𝐴𝐵 ∧ ¬ (𝐹𝐴) = ∅) → (𝐹𝐴) ∈ (𝐹𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  ∅c0 3874   class class class wbr 4583   “ cima 5041  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fv 5812 This theorem is referenced by:  ovima0  6711  setrec2fun  42238
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