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Theorem fvmptdf 5285
Description: Alternate deduction version of fvmpt 5276, 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 1437 . 2 𝑥𝜑
2 fvmptdf.4 . . . 4 𝑥𝐹
3 nfmpt1 3877 . . . 4 𝑥(𝑥𝐷𝐵)
42, 3nfeq 2201 . . 3 𝑥 𝐹 = (𝑥𝐷𝐵)
5 fvmptdf.5 . . 3 𝑥𝜓
64, 5nfim 1480 . 2 𝑥(𝐹 = (𝑥𝐷𝐵) → 𝜓)
7 fvmptdf.1 . . . 4 (𝜑𝐴𝐷)
8 elex 2583 . . . 4 (𝐴𝐷𝐴 ∈ V)
97, 8syl 14 . . 3 (𝜑𝐴 ∈ V)
10 isset 2578 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
119, 10sylib 131 . 2 (𝜑 → ∃𝑥 𝑥 = 𝐴)
12 fveq1 5204 . . 3 (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴))
13 simpr 107 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐴)
1413fveq2d 5209 . . . . . 6 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝑥) = ((𝑥𝐷𝐵)‘𝐴))
157adantr 265 . . . . . . . 8 ((𝜑𝑥 = 𝐴) → 𝐴𝐷)
1613, 15eqeltrd 2130 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥𝐷)
17 fvmptdf.2 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝐵𝑉)
18 eqid 2056 . . . . . . . 8 (𝑥𝐷𝐵) = (𝑥𝐷𝐵)
1918fvmpt2 5281 . . . . . . 7 ((𝑥𝐷𝐵𝑉) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
2016, 17, 19syl2anc 397 . . . . . 6 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
2114, 20eqtr3d 2090 . . . . 5 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝐴) = 𝐵)
2221eqeq2d 2067 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴) ↔ (𝐹𝐴) = 𝐵))
23 fvmptdf.3 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))
2422, 23sylbid 143 . . 3 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴) → 𝜓))
2512, 24syl5 32 . 2 ((𝜑𝑥 = 𝐴) → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
261, 6, 11, 25exlimdd 1768 1 (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wnf 1365  wex 1397  wcel 1409  wnfc 2181  Vcvv 2574  cmpt 3845  cfv 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-csb 2880  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-iota 4894  df-fun 4931  df-fv 4937
This theorem is referenced by:  fvmptdv  5286
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