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Theorem rexrn 5247
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 5135 . . 3 ((Fun 𝐹 y dom 𝐹) → (𝐹y) V)
21funfni 4942 . 2 ((𝐹 Fn A y A) → (𝐹y) V)
3 fvelrnb 5164 . . 3 (𝐹 Fn A → (x ran 𝐹y A (𝐹y) = x))
4 eqcom 2039 . . . 4 ((𝐹y) = xx = (𝐹y))
54rexbii 2325 . . 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 4163 1 (𝐹 Fn A → (x ran 𝐹φy A ψ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242   wcel 1390  wrex 2301  Vcvv 2551  ran crn 4289   Fn wfn 4840  cfv 4845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  elrnrexdm  5249  rexrnmpt  5253  cbvexfo  5369
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