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Theorem rexima 7231
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 6897 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝑦𝐵) → (𝐹𝑦) ∈ V)
2 fvelimab 6955 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑥 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 (𝐹𝑦) = 𝑥))
3 eqcom 2731 . . . 4 ((𝐹𝑦) = 𝑥𝑥 = (𝐹𝑦))
43rexbii 3086 . . 3 (∃𝑦𝐵 (𝐹𝑦) = 𝑥 ↔ ∃𝑦𝐵 𝑥 = (𝐹𝑦))
52, 4bitrdi 287 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑥 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 𝑥 = (𝐹𝑦)))
6 rexima.x . . 3 (𝑥 = (𝐹𝑦) → (𝜑𝜓))
76adantl 481 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝑥 = (𝐹𝑦)) → (𝜑𝜓))
81, 5, 7rexxfr2d 5400 1 ((𝐹 Fn 𝐴𝐵𝐴) → (∃𝑥 ∈ (𝐹𝐵)𝜑 ↔ ∃𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wrex 3062  Vcvv 3466  wss 3941  cima 5670   Fn wfn 6529  cfv 6534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-fv 6542
This theorem is referenced by:  supisolem  9465  ipodrsima  18502  lmflf  23853  caucfil  25155  dyadmbllem  25472  lhop1lem  25890  nummin  34613  mblfinlem1  37029  itg2gt0cn  37047
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