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Theorem fvmptd 5196
Description: Deduction version of fvmpt 5192. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
fvmptd.1 (φ𝐹 = (x 𝐷B))
fvmptd.2 ((φ x = A) → B = 𝐶)
fvmptd.3 (φA 𝐷)
fvmptd.4 (φ𝐶 𝑉)
Assertion
Ref Expression
fvmptd (φ → (𝐹A) = 𝐶)
Distinct variable groups:   x,A   x,𝐶   x,𝐷   φ,x
Allowed substitution hints:   B(x)   𝐹(x)   𝑉(x)

Proof of Theorem fvmptd
StepHypRef Expression
1 fvmptd.1 . . 3 (φ𝐹 = (x 𝐷B))
21fveq1d 5123 . 2 (φ → (𝐹A) = ((x 𝐷B)‘A))
3 fvmptd.3 . . 3 (φA 𝐷)
4 fvmptd.2 . . . . 5 ((φ x = A) → B = 𝐶)
53, 4csbied 2886 . . . 4 (φA / xB = 𝐶)
6 fvmptd.4 . . . 4 (φ𝐶 𝑉)
75, 6eqeltrd 2111 . . 3 (φA / xB 𝑉)
8 eqid 2037 . . . 4 (x 𝐷B) = (x 𝐷B)
98fvmpts 5193 . . 3 ((A 𝐷 A / xB 𝑉) → ((x 𝐷B)‘A) = A / xB)
103, 7, 9syl2anc 391 . 2 (φ → ((x 𝐷B)‘A) = A / xB)
112, 10, 53eqtrd 2073 1 (φ → (𝐹A) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  csb 2846  cmpt 3809  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-csb 2847  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-iota 4810  df-fun 4847  df-fv 4853
This theorem is referenced by:  fvmptdv2  5203  rdgivallem  5908
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