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Theorem fvmptdv2 7022
Description: Alternate deduction version of fvmpt 7004, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
fvmptdv2.1 (𝜑𝐴𝐷)
fvmptdv2.2 ((𝜑𝑥 = 𝐴) → 𝐵𝑉)
fvmptdv2.3 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
Assertion
Ref Expression
fvmptdv2 (𝜑 → (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = 𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fvmptdv2
StepHypRef Expression
1 eqidd 2726 . . 3 (𝜑 → (𝑥𝐷𝐵) = (𝑥𝐷𝐵))
2 fvmptdv2.3 . . 3 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
3 fvmptdv2.1 . . 3 (𝜑𝐴𝐷)
43elexd 3483 . . . . 5 (𝜑𝐴 ∈ V)
5 isset 3474 . . . . 5 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
64, 5sylib 217 . . . 4 (𝜑 → ∃𝑥 𝑥 = 𝐴)
7 fvmptdv2.2 . . . . . 6 ((𝜑𝑥 = 𝐴) → 𝐵𝑉)
87elexd 3483 . . . . 5 ((𝜑𝑥 = 𝐴) → 𝐵 ∈ V)
92, 8eqeltrrd 2826 . . . 4 ((𝜑𝑥 = 𝐴) → 𝐶 ∈ V)
106, 9exlimddv 1930 . . 3 (𝜑𝐶 ∈ V)
111, 2, 3, 10fvmptd 7011 . 2 (𝜑 → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
12 fveq1 6895 . . 3 (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴))
1312eqeq1d 2727 . 2 (𝐹 = (𝑥𝐷𝐵) → ((𝐹𝐴) = 𝐶 ↔ ((𝑥𝐷𝐵)‘𝐴) = 𝐶))
1411, 13syl5ibrcom 246 1 (𝜑 → (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wex 1773  wcel 2098  Vcvv 3461  cmpt 5232  cfv 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557
This theorem is referenced by:  curf12  18222  curf2  18224  yonedalem4b  18271
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