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Theorem rexima 5319
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 A BA) → (x (𝐹B)φy B 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 ssel2 2917 . . . 4 ((BA y B) → y A)
2 funfvex 5117 . . . . 5 ((Fun 𝐹 y dom 𝐹) → (𝐹y) V)
32funfni 4925 . . . 4 ((𝐹 Fn A y A) → (𝐹y) V)
41, 3sylan2 270 . . 3 ((𝐹 Fn A (BA y B)) → (𝐹y) V)
54anassrs 382 . 2 (((𝐹 Fn A BA) y B) → (𝐹y) V)
6 fvelimab 5154 . . 3 ((𝐹 Fn A BA) → (x (𝐹B) ↔ y B (𝐹y) = x))
7 eqcom 2024 . . . 4 ((𝐹y) = xx = (𝐹y))
87rexbii 2309 . . 3 (y B (𝐹y) = xy B x = (𝐹y))
96, 8syl6bb 185 . 2 ((𝐹 Fn A BA) → (x (𝐹B) ↔ y B x = (𝐹y)))
10 rexima.x . . 3 (x = (𝐹y) → (φψ))
1110adantl 262 . 2 (((𝐹 Fn A BA) x = (𝐹y)) → (φψ))
125, 9, 11rexxfr2d 4147 1 ((𝐹 Fn A BA) → (x (𝐹B)φy B ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228   wcel 1374  wrex 2285  Vcvv 2535  wss 2894  cima 4275   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-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837
This theorem is referenced by: (None)
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