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Theorem fvmptdf 5258
 Description: Alternate deduction version of fvmpt 5249, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
fvmptdf.1 (𝜑𝐴𝐷)
fvmptdf.2 ((𝜑𝑥 = 𝐴) → 𝐵𝑉)
fvmptdf.3 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))
fvmptdf.4 𝑥𝐹
fvmptdf.5 𝑥𝜓
Assertion
Ref Expression
fvmptdf (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 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 1421 . 2 𝑥𝜑
2 fvmptdf.4 . . . 4 𝑥𝐹
3 nfmpt1 3850 . . . 4 𝑥(𝑥𝐷𝐵)
42, 3nfeq 2185 . . 3 𝑥 𝐹 = (𝑥𝐷𝐵)
5 fvmptdf.5 . . 3 𝑥𝜓
64, 5nfim 1464 . 2 𝑥(𝐹 = (𝑥𝐷𝐵) → 𝜓)
7 fvmptdf.1 . . . 4 (𝜑𝐴𝐷)
8 elex 2566 . . . 4 (𝐴𝐷𝐴 ∈ V)
97, 8syl 14 . . 3 (𝜑𝐴 ∈ V)
10 isset 2561 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
119, 10sylib 127 . 2 (𝜑 → ∃𝑥 𝑥 = 𝐴)
12 fveq1 5177 . . 3 (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴))
13 simpr 103 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐴)
1413fveq2d 5182 . . . . . 6 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝑥) = ((𝑥𝐷𝐵)‘𝐴))
157adantr 261 . . . . . . . 8 ((𝜑𝑥 = 𝐴) → 𝐴𝐷)
1613, 15eqeltrd 2114 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥𝐷)
17 fvmptdf.2 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝐵𝑉)
18 eqid 2040 . . . . . . . 8 (𝑥𝐷𝐵) = (𝑥𝐷𝐵)
1918fvmpt2 5254 . . . . . . 7 ((𝑥𝐷𝐵𝑉) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
2016, 17, 19syl2anc 391 . . . . . 6 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
2114, 20eqtr3d 2074 . . . . 5 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝐴) = 𝐵)
2221eqeq2d 2051 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴) ↔ (𝐹𝐴) = 𝐵))
23 fvmptdf.3 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))
2422, 23sylbid 139 . . 3 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴) → 𝜓))
2512, 24syl5 28 . 2 ((𝜑𝑥 = 𝐴) → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
261, 6, 11, 25exlimdd 1752 1 (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  Ⅎwnf 1349  ∃wex 1381   ∈ wcel 1393  Ⅎwnfc 2165  Vcvv 2557   ↦ cmpt 3818  ‘cfv 4902 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910 This theorem is referenced by:  fvmptdv  5259
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