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Theorem fvmptdv2 6206
 Description: Alternate deduction version of fvmpt 6191, 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 2611 . . 3 (𝜑 → (𝑥𝐷𝐵) = (𝑥𝐷𝐵))
2 fvmptdv2.3 . . 3 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
3 fvmptdv2.1 . . 3 (𝜑𝐴𝐷)
4 elex 3185 . . . . . 6 (𝐴𝐷𝐴 ∈ V)
53, 4syl 17 . . . . 5 (𝜑𝐴 ∈ V)
6 isset 3180 . . . . 5 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
75, 6sylib 207 . . . 4 (𝜑 → ∃𝑥 𝑥 = 𝐴)
8 fvmptdv2.2 . . . . . 6 ((𝜑𝑥 = 𝐴) → 𝐵𝑉)
9 elex 3185 . . . . . 6 (𝐵𝑉𝐵 ∈ V)
108, 9syl 17 . . . . 5 ((𝜑𝑥 = 𝐴) → 𝐵 ∈ V)
112, 10eqeltrrd 2689 . . . 4 ((𝜑𝑥 = 𝐴) → 𝐶 ∈ V)
127, 11exlimddv 1850 . . 3 (𝜑𝐶 ∈ V)
131, 2, 3, 12fvmptd 6197 . 2 (𝜑 → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
14 fveq1 6102 . . 3 (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴))
1514eqeq1d 2612 . 2 (𝐹 = (𝑥𝐷𝐵) → ((𝐹𝐴) = 𝐶 ↔ ((𝑥𝐷𝐵)‘𝐴) = 𝐶))
1613, 15syl5ibrcom 236 1 (𝜑 → (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = 𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   ↦ cmpt 4643  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812 This theorem is referenced by:  curf12  16690  curf2  16692  yonedalem4b  16739
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