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Theorem rexrn 5356
Description: Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
Hypothesis
Ref Expression
rexrn.1 (𝑥 = (𝐹𝑦) → (𝜑𝜓))
Assertion
Ref Expression
rexrn (𝐹 Fn 𝐴 → (∃𝑥 ∈ ran 𝐹𝜑 ↔ ∃𝑦𝐴 𝜓))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦   𝜓,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem rexrn
StepHypRef Expression
1 funfvex 5243 . . 3 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝐹𝑦) ∈ V)
21funfni 5050 . 2 ((𝐹 Fn 𝐴𝑦𝐴) → (𝐹𝑦) ∈ V)
3 fvelrnb 5273 . . 3 (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦𝐴 (𝐹𝑦) = 𝑥))
4 eqcom 2085 . . . 4 ((𝐹𝑦) = 𝑥𝑥 = (𝐹𝑦))
54rexbii 2378 . . 3 (∃𝑦𝐴 (𝐹𝑦) = 𝑥 ↔ ∃𝑦𝐴 𝑥 = (𝐹𝑦))
63, 5syl6bb 194 . 2 (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦𝐴 𝑥 = (𝐹𝑦)))
7 rexrn.1 . . 3 (𝑥 = (𝐹𝑦) → (𝜑𝜓))
87adantl 271 . 2 ((𝐹 Fn 𝐴𝑥 = (𝐹𝑦)) → (𝜑𝜓))
92, 6, 8rexxfr2d 4243 1 (𝐹 Fn 𝐴 → (∃𝑥 ∈ ran 𝐹𝜑 ↔ ∃𝑦𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wcel 1434  wrex 2354  Vcvv 2610  ran crn 4392   Fn wfn 4947  cfv 4952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-iota 4917  df-fun 4954  df-fn 4955  df-fv 4960
This theorem is referenced by:  elrnrexdm  5358  rexrnmpt  5362  cbvexfo  5477  rexanuz  10075
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