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Theorem fvmptdf 6204
 Description: Alternate deduction version of fvmpt 6191, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
fvmptdf.1 (𝜑𝐴𝐷)
fvmptdf.2 ((𝜑𝑥 = 𝐴) → 𝐵𝑉)
fvmptdf.3 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))
fvmptdf.4 𝑥𝐹
fvmptdf.5 𝑥𝜓
Assertion
Ref Expression
fvmptdf (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐷   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fvmptdf
StepHypRef Expression
1 nfv 1830 . 2 𝑥𝜑
2 fvmptdf.4 . . . 4 𝑥𝐹
3 nfmpt1 4675 . . . 4 𝑥(𝑥𝐷𝐵)
42, 3nfeq 2762 . . 3 𝑥 𝐹 = (𝑥𝐷𝐵)
5 fvmptdf.5 . . 3 𝑥𝜓
64, 5nfim 1813 . 2 𝑥(𝐹 = (𝑥𝐷𝐵) → 𝜓)
7 fvmptdf.1 . . . 4 (𝜑𝐴𝐷)
87elexd 3187 . . 3 (𝜑𝐴 ∈ V)
9 isset 3180 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
108, 9sylib 207 . 2 (𝜑 → ∃𝑥 𝑥 = 𝐴)
11 fveq1 6102 . . 3 (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴))
12 simpr 476 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐴)
1312fveq2d 6107 . . . . . 6 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝑥) = ((𝑥𝐷𝐵)‘𝐴))
147adantr 480 . . . . . . . 8 ((𝜑𝑥 = 𝐴) → 𝐴𝐷)
1512, 14eqeltrd 2688 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥𝐷)
16 fvmptdf.2 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝐵𝑉)
17 eqid 2610 . . . . . . . 8 (𝑥𝐷𝐵) = (𝑥𝐷𝐵)
1817fvmpt2 6200 . . . . . . 7 ((𝑥𝐷𝐵𝑉) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
1915, 16, 18syl2anc 691 . . . . . 6 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
2013, 19eqtr3d 2646 . . . . 5 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝐴) = 𝐵)
2120eqeq2d 2620 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴) ↔ (𝐹𝐴) = 𝐵))
22 fvmptdf.3 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))
2321, 22sylbid 229 . . 3 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴) → 𝜓))
2411, 23syl5 33 . 2 ((𝜑𝑥 = 𝐴) → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
251, 6, 10, 24exlimdd 2075 1 (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738  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-8 1979  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-pow 4769  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-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812 This theorem is referenced by:  fvmptdv  6205  yonedalem4b  16739
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