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Theorem fvmptdv2 7034
Description: Alternate deduction version of fvmpt 7016, 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 2738 . . 3 (𝜑 → (𝑥𝐷𝐵) = (𝑥𝐷𝐵))
2 fvmptdv2.3 . . 3 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
3 fvmptdv2.1 . . 3 (𝜑𝐴𝐷)
43elexd 3504 . . . . 5 (𝜑𝐴 ∈ V)
5 isset 3494 . . . . 5 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
64, 5sylib 218 . . . 4 (𝜑 → ∃𝑥 𝑥 = 𝐴)
7 fvmptdv2.2 . . . . . 6 ((𝜑𝑥 = 𝐴) → 𝐵𝑉)
87elexd 3504 . . . . 5 ((𝜑𝑥 = 𝐴) → 𝐵 ∈ V)
92, 8eqeltrrd 2842 . . . 4 ((𝜑𝑥 = 𝐴) → 𝐶 ∈ V)
106, 9exlimddv 1935 . . 3 (𝜑𝐶 ∈ V)
111, 2, 3, 10fvmptd 7023 . 2 (𝜑 → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
12 fveq1 6905 . . 3 (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴))
1312eqeq1d 2739 . 2 (𝐹 = (𝑥𝐷𝐵) → ((𝐹𝐴) = 𝐶 ↔ ((𝑥𝐷𝐵)‘𝐴) = 𝐶))
1411, 13syl5ibrcom 247 1 (𝜑 → (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  cmpt 5225  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569
This theorem is referenced by:  curf12  18272  curf2  18274  yonedalem4b  18321
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