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Theorem fvmptdf 5624
Description: Alternate deduction version of fvmpt 5614, 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 1539 . 2 𝑥𝜑
2 fvmptdf.4 . . . 4 𝑥𝐹
3 nfmpt1 4111 . . . 4 𝑥(𝑥𝐷𝐵)
42, 3nfeq 2340 . . 3 𝑥 𝐹 = (𝑥𝐷𝐵)
5 fvmptdf.5 . . 3 𝑥𝜓
64, 5nfim 1583 . 2 𝑥(𝐹 = (𝑥𝐷𝐵) → 𝜓)
7 fvmptdf.1 . . . 4 (𝜑𝐴𝐷)
8 elex 2763 . . . 4 (𝐴𝐷𝐴 ∈ V)
97, 8syl 14 . . 3 (𝜑𝐴 ∈ V)
10 isset 2758 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
119, 10sylib 122 . 2 (𝜑 → ∃𝑥 𝑥 = 𝐴)
12 fveq1 5533 . . 3 (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴))
13 simpr 110 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐴)
1413fveq2d 5538 . . . . . 6 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝑥) = ((𝑥𝐷𝐵)‘𝐴))
157adantr 276 . . . . . . . 8 ((𝜑𝑥 = 𝐴) → 𝐴𝐷)
1613, 15eqeltrd 2266 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥𝐷)
17 fvmptdf.2 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝐵𝑉)
18 eqid 2189 . . . . . . . 8 (𝑥𝐷𝐵) = (𝑥𝐷𝐵)
1918fvmpt2 5620 . . . . . . 7 ((𝑥𝐷𝐵𝑉) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
2016, 17, 19syl2anc 411 . . . . . 6 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
2114, 20eqtr3d 2224 . . . . 5 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝐴) = 𝐵)
2221eqeq2d 2201 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴) ↔ (𝐹𝐴) = 𝐵))
23 fvmptdf.3 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))
2422, 23sylbid 150 . . 3 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴) → 𝜓))
2512, 24syl5 32 . 2 ((𝜑𝑥 = 𝐴) → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
261, 6, 11, 25exlimdd 1883 1 (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wnf 1471  wex 1503  wcel 2160  wnfc 2319  Vcvv 2752  cmpt 4079  cfv 5235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243
This theorem is referenced by:  fvmptdv  5625
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