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Theorem rexima 6378
Description: Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
Hypothesis
Ref Expression
rexima.x (𝑥 = (𝐹𝑦) → (𝜑𝜓))
Assertion
Ref Expression
rexima ((𝐹 Fn 𝐴𝐵𝐴) → (∃𝑥 ∈ (𝐹𝐵)𝜑 ↔ ∃𝑦𝐵 𝜓))
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝐹,𝑦   𝑥,𝐵,𝑦   𝑥,𝐴,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem rexima
StepHypRef Expression
1 fvex 6097 . . 3 (𝐹𝑦) ∈ V
21a1i 11 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝑦𝐵) → (𝐹𝑦) ∈ V)
3 fvelimab 6147 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑥 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 (𝐹𝑦) = 𝑥))
4 eqcom 2616 . . . 4 ((𝐹𝑦) = 𝑥𝑥 = (𝐹𝑦))
54rexbii 3022 . . 3 (∃𝑦𝐵 (𝐹𝑦) = 𝑥 ↔ ∃𝑦𝐵 𝑥 = (𝐹𝑦))
63, 5syl6bb 274 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑥 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
7 rexima.x . . 3 (𝑥 = (𝐹𝑦) → (𝜑𝜓))
87adantl 480 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝑥 = (𝐹𝑦)) → (𝜑𝜓))
92, 6, 8rexxfr2d 4803 1 ((𝐹 Fn 𝐴𝐵𝐴) → (∃𝑥 ∈ (𝐹𝐵)𝜑 ↔ ∃𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wrex 2896  Vcvv 3172  wss 3539  cima 5030   Fn wfn 5784  cfv 5789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-fv 5797
This theorem is referenced by:  supisolem  8239  ipodrsima  16936  lmflf  21566  caucfil  22833  dyadmbllem  23117  lhop1lem  23524  mblfinlem1  32399  itg2gt0cn  32418
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