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Theorem fvmptdf 5183
Description: Alternate deduction version of fvmpt 5174, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
fvmptdf.1 (φA 𝐷)
fvmptdf.2 ((φ x = A) → B 𝑉)
fvmptdf.3 ((φ x = A) → ((𝐹A) = Bψ))
fvmptdf.4 x𝐹
fvmptdf.5 xψ
Assertion
Ref Expression
fvmptdf (φ → (𝐹 = (x 𝐷B) → ψ))
Distinct variable groups:   x,A   x,𝐷   φ,x
Allowed substitution hints:   ψ(x)   B(x)   𝐹(x)   𝑉(x)

Proof of Theorem fvmptdf
StepHypRef Expression
1 nfv 1402 . 2 xφ
2 fvmptdf.4 . . . 4 x𝐹
3 nfmpt1 3824 . . . 4 x(x 𝐷B)
42, 3nfeq 2167 . . 3 x 𝐹 = (x 𝐷B)
5 fvmptdf.5 . . 3 xψ
64, 5nfim 1446 . 2 x(𝐹 = (x 𝐷B) → ψ)
7 fvmptdf.1 . . . 4 (φA 𝐷)
8 elex 2543 . . . 4 (A 𝐷A V)
97, 8syl 14 . . 3 (φA V)
10 isset 2539 . . 3 (A V ↔ x x = A)
119, 10sylib 127 . 2 (φx x = A)
12 fveq1 5102 . . 3 (𝐹 = (x 𝐷B) → (𝐹A) = ((x 𝐷B)‘A))
13 simpr 103 . . . . . . 7 ((φ x = A) → x = A)
1413fveq2d 5107 . . . . . 6 ((φ x = A) → ((x 𝐷B)‘x) = ((x 𝐷B)‘A))
157adantr 261 . . . . . . . 8 ((φ x = A) → A 𝐷)
1613, 15eqeltrd 2096 . . . . . . 7 ((φ x = A) → x 𝐷)
17 fvmptdf.2 . . . . . . 7 ((φ x = A) → B 𝑉)
18 eqid 2022 . . . . . . . 8 (x 𝐷B) = (x 𝐷B)
1918fvmpt2 5179 . . . . . . 7 ((x 𝐷 B 𝑉) → ((x 𝐷B)‘x) = B)
2016, 17, 19syl2anc 393 . . . . . 6 ((φ x = A) → ((x 𝐷B)‘x) = B)
2114, 20eqtr3d 2056 . . . . 5 ((φ x = A) → ((x 𝐷B)‘A) = B)
2221eqeq2d 2033 . . . 4 ((φ x = A) → ((𝐹A) = ((x 𝐷B)‘A) ↔ (𝐹A) = B))
23 fvmptdf.3 . . . 4 ((φ x = A) → ((𝐹A) = Bψ))
2422, 23sylbid 139 . . 3 ((φ x = A) → ((𝐹A) = ((x 𝐷B)‘A) → ψ))
2512, 24syl5 28 . 2 ((φ x = A) → (𝐹 = (x 𝐷B) → ψ))
261, 6, 11, 25exlimdd 1734 1 (φ → (𝐹 = (x 𝐷B) → ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228  wnf 1329  wex 1362   wcel 1374  wnfc 2147  Vcvv 2535  cmpt 3792  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-csb 2830  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-iota 4794  df-fun 4831  df-fv 4837
This theorem is referenced by:  fvmptdv  5184
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