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Theorem fvmptdv2 6610
Description: Alternate deduction version of fvmpt 6593, 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 2772 . . 3 (𝜑 → (𝑥𝐷𝐵) = (𝑥𝐷𝐵))
2 fvmptdv2.3 . . 3 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
3 fvmptdv2.1 . . 3 (𝜑𝐴𝐷)
43elexd 3428 . . . . 5 (𝜑𝐴 ∈ V)
5 isset 3420 . . . . 5 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
64, 5sylib 210 . . . 4 (𝜑 → ∃𝑥 𝑥 = 𝐴)
7 fvmptdv2.2 . . . . . 6 ((𝜑𝑥 = 𝐴) → 𝐵𝑉)
87elexd 3428 . . . . 5 ((𝜑𝑥 = 𝐴) → 𝐵 ∈ V)
92, 8eqeltrrd 2860 . . . 4 ((𝜑𝑥 = 𝐴) → 𝐶 ∈ V)
106, 9exlimddv 1895 . . 3 (𝜑𝐶 ∈ V)
111, 2, 3, 10fvmptd 6599 . 2 (𝜑 → ((𝑥𝐷𝐵)‘𝐴) = 𝐶)
12 fveq1 6495 . . 3 (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴))
1312eqeq1d 2773 . 2 (𝐹 = (𝑥𝐷𝐵) → ((𝐹𝐴) = 𝐶 ↔ ((𝑥𝐷𝐵)‘𝐴) = 𝐶))
1411, 13syl5ibrcom 239 1 (𝜑 → (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1508  wex 1743  wcel 2051  Vcvv 3408  cmpt 5004  cfv 6185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-iota 6149  df-fun 6187  df-fv 6193
This theorem is referenced by:  curf12  17347  curf2  17349  yonedalem4b  17396
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