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Theorem fvmptdf 5201
Description: Alternate deduction version of fvmpt 5192, 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 1418 . 2 xφ
2 fvmptdf.4 . . . 4 x𝐹
3 nfmpt1 3841 . . . 4 x(x 𝐷B)
42, 3nfeq 2182 . . 3 x 𝐹 = (x 𝐷B)
5 fvmptdf.5 . . 3 xψ
64, 5nfim 1461 . 2 x(𝐹 = (x 𝐷B) → ψ)
7 fvmptdf.1 . . . 4 (φA 𝐷)
8 elex 2560 . . . 4 (A 𝐷A V)
97, 8syl 14 . . 3 (φA V)
10 isset 2555 . . 3 (A V ↔ x x = A)
119, 10sylib 127 . 2 (φx x = A)
12 fveq1 5120 . . 3 (𝐹 = (x 𝐷B) → (𝐹A) = ((x 𝐷B)‘A))
13 simpr 103 . . . . . . 7 ((φ x = A) → x = A)
1413fveq2d 5125 . . . . . 6 ((φ x = A) → ((x 𝐷B)‘x) = ((x 𝐷B)‘A))
157adantr 261 . . . . . . . 8 ((φ x = A) → A 𝐷)
1613, 15eqeltrd 2111 . . . . . . 7 ((φ x = A) → x 𝐷)
17 fvmptdf.2 . . . . . . 7 ((φ x = A) → B 𝑉)
18 eqid 2037 . . . . . . . 8 (x 𝐷B) = (x 𝐷B)
1918fvmpt2 5197 . . . . . . 7 ((x 𝐷 B 𝑉) → ((x 𝐷B)‘x) = B)
2016, 17, 19syl2anc 391 . . . . . 6 ((φ x = A) → ((x 𝐷B)‘x) = B)
2114, 20eqtr3d 2071 . . . . 5 ((φ x = A) → ((x 𝐷B)‘A) = B)
2221eqeq2d 2048 . . . 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 1749 1 (φ → (𝐹 = (x 𝐷B) → ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wnf 1346  wex 1378   wcel 1390  wnfc 2162  Vcvv 2551  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-bndl 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:  fvmptdv  5202
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