MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvmptdv2 Structured version   Visualization version   GIF version

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 2736 . . 3 (𝜑 → (𝑥𝐷𝐵) = (𝑥𝐷𝐵))
2 fvmptdv2.3 . . 3 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
3 fvmptdv2.1 . . 3 (𝜑𝐴𝐷)
43elexd 3502 . . . . 5 (𝜑𝐴 ∈ V)
5 isset 3492 . . . . 5 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
64, 5sylib 218 . . . 4 (𝜑 → ∃𝑥 𝑥 = 𝐴)
7 fvmptdv2.2 . . . . . 6 ((𝜑𝑥 = 𝐴) → 𝐵𝑉)
87elexd 3502 . . . . 5 ((𝜑𝑥 = 𝐴) → 𝐵 ∈ V)
92, 8eqeltrrd 2840 . . . 4 ((𝜑𝑥 = 𝐴) → 𝐶 ∈ V)
106, 9exlimddv 1933 . . 3 (𝜑𝐶 ∈ V)
111, 2, 3, 10fvmptd 7023 . 2 (𝜑 → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
12 fveq1 6906 . . 3 (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴))
1312eqeq1d 2737 . 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 1537  wex 1776  wcel 2106  Vcvv 3478  cmpt 5231  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571
This theorem is referenced by:  curf12  18284  curf2  18286  yonedalem4b  18333
  Copyright terms: Public domain W3C validator