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Theorem fvmptdf 5508
Description: Alternate deduction version of fvmpt 5498, 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 1508 . 2 𝑥𝜑
2 fvmptdf.4 . . . 4 𝑥𝐹
3 nfmpt1 4021 . . . 4 𝑥(𝑥𝐷𝐵)
42, 3nfeq 2289 . . 3 𝑥 𝐹 = (𝑥𝐷𝐵)
5 fvmptdf.5 . . 3 𝑥𝜓
64, 5nfim 1551 . 2 𝑥(𝐹 = (𝑥𝐷𝐵) → 𝜓)
7 fvmptdf.1 . . . 4 (𝜑𝐴𝐷)
8 elex 2697 . . . 4 (𝐴𝐷𝐴 ∈ V)
97, 8syl 14 . . 3 (𝜑𝐴 ∈ V)
10 isset 2692 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
119, 10sylib 121 . 2 (𝜑 → ∃𝑥 𝑥 = 𝐴)
12 fveq1 5420 . . 3 (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴))
13 simpr 109 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐴)
1413fveq2d 5425 . . . . . 6 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝑥) = ((𝑥𝐷𝐵)‘𝐴))
157adantr 274 . . . . . . . 8 ((𝜑𝑥 = 𝐴) → 𝐴𝐷)
1613, 15eqeltrd 2216 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥𝐷)
17 fvmptdf.2 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝐵𝑉)
18 eqid 2139 . . . . . . . 8 (𝑥𝐷𝐵) = (𝑥𝐷𝐵)
1918fvmpt2 5504 . . . . . . 7 ((𝑥𝐷𝐵𝑉) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
2016, 17, 19syl2anc 408 . . . . . 6 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
2114, 20eqtr3d 2174 . . . . 5 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝐴) = 𝐵)
2221eqeq2d 2151 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴) ↔ (𝐹𝐴) = 𝐵))
23 fvmptdf.3 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))
2422, 23sylbid 149 . . 3 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴) → 𝜓))
2512, 24syl5 32 . 2 ((𝜑𝑥 = 𝐴) → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
261, 6, 11, 25exlimdd 1844 1 (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wnf 1436  wex 1468  wcel 1480  wnfc 2268  Vcvv 2686  cmpt 3989  cfv 5123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131
This theorem is referenced by:  fvmptdv  5509
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