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Theorem rexrn 5229
 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 𝐹φy A 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 5117 . . 3 ((Fun 𝐹 y dom 𝐹) → (𝐹y) V)
21funfni 4925 . 2 ((𝐹 Fn A y A) → (𝐹y) V)
3 fvelrnb 5146 . . 3 (𝐹 Fn A → (x ran 𝐹y A (𝐹y) = x))
4 eqcom 2024 . . . 4 ((𝐹y) = xx = (𝐹y))
54rexbii 2309 . . 3 (y A (𝐹y) = xy A x = (𝐹y))
63, 5syl6bb 185 . 2 (𝐹 Fn A → (x ran 𝐹y A x = (𝐹y)))
7 rexrn.1 . . 3 (x = (𝐹y) → (φψ))
87adantl 262 . 2 ((𝐹 Fn A x = (𝐹y)) → (φψ))
92, 6, 8rexxfr2d 4147 1 (𝐹 Fn A → (x ran 𝐹φy A ψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1228   ∈ wcel 1374  ∃wrex 2285  Vcvv 2535  ran crn 4273   Fn wfn 4824  ‘cfv 4829 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837 This theorem is referenced by:  elrnrexdm  5231  rexrnmpt  5235  cbvexfo  5351
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