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Theorem rexima 6824
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 fvexd 6514 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝑦𝐵) → (𝐹𝑦) ∈ V)
2 fvelimab 6566 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑥 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 (𝐹𝑦) = 𝑥))
3 eqcom 2786 . . . 4 ((𝐹𝑦) = 𝑥𝑥 = (𝐹𝑦))
43rexbii 3195 . . 3 (∃𝑦𝐵 (𝐹𝑦) = 𝑥 ↔ ∃𝑦𝐵 𝑥 = (𝐹𝑦))
52, 4syl6bb 279 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑥 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
6 rexima.x . . 3 (𝑥 = (𝐹𝑦) → (𝜑𝜓))
76adantl 474 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝑥 = (𝐹𝑦)) → (𝜑𝜓))
81, 5, 7rexxfr2d 5165 1 ((𝐹 Fn 𝐴𝐵𝐴) → (∃𝑥 ∈ (𝐹𝐵)𝜑 ↔ ∃𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2050  wrex 3090  Vcvv 3416  wss 3830  cima 5410   Fn wfn 6183  cfv 6188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-fv 6196
This theorem is referenced by:  supisolem  8732  ipodrsima  17633  lmflf  22317  caucfil  23589  dyadmbllem  23903  lhop1lem  24313  mblfinlem1  34367  itg2gt0cn  34385
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