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Theorem eliman0 6221
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 6213 . . . . 5 (𝐴𝐹(𝐹𝐴) ∨ (𝐹𝐴) = ∅)
2 orcom 402 . . . . 5 ((𝐴𝐹(𝐹𝐴) ∨ (𝐹𝐴) = ∅) ↔ ((𝐹𝐴) = ∅ ∨ 𝐴𝐹(𝐹𝐴)))
31, 2mpbi 220 . . . 4 ((𝐹𝐴) = ∅ ∨ 𝐴𝐹(𝐹𝐴))
43ori 390 . . 3 (¬ (𝐹𝐴) = ∅ → 𝐴𝐹(𝐹𝐴))
5 breq1 4654 . . . 4 (𝑥 = 𝐴 → (𝑥𝐹(𝐹𝐴) ↔ 𝐴𝐹(𝐹𝐴)))
65rspcev 3307 . . 3 ((𝐴𝐵𝐴𝐹(𝐹𝐴)) → ∃𝑥𝐵 𝑥𝐹(𝐹𝐴))
74, 6sylan2 491 . 2 ((𝐴𝐵 ∧ ¬ (𝐹𝐴) = ∅) → ∃𝑥𝐵 𝑥𝐹(𝐹𝐴))
8 fvex 6199 . . 3 (𝐹𝐴) ∈ V
98elima 5469 . 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  wo 383  wa 384   = wceq 1482  wcel 1989  wrex 2912  c0 3913   class class class wbr 4651  cima 5115  cfv 5886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-xp 5118  df-cnv 5120  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fv 5894
This theorem is referenced by:  ovima0  6810  setrec2fun  42210
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