Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvmptd3f Structured version   Visualization version   GIF version

Theorem fvmptd3f 6437
 Description: Alternate deduction version of fvmpt 6424 with three non-freeness hypotheses instead of distinct variable conditions. (Contributed by AV, 19-Jan-2022.)
Hypotheses
Ref Expression
fvmptdf.1 (𝜑𝐴𝐷)
fvmptdf.2 ((𝜑𝑥 = 𝐴) → 𝐵𝑉)
fvmptdf.3 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))
fvmptd3f.4 𝑥𝐹
fvmptd3f.5 𝑥𝜓
fvmptd3f.6 𝑥𝜑
Assertion
Ref Expression
fvmptd3f (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐷
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fvmptd3f
StepHypRef Expression
1 fvmptd3f.6 . 2 𝑥𝜑
2 fvmptd3f.4 . . . 4 𝑥𝐹
3 nfmpt1 4881 . . . 4 𝑥(𝑥𝐷𝐵)
42, 3nfeq 2925 . . 3 𝑥 𝐹 = (𝑥𝐷𝐵)
5 fvmptd3f.5 . . 3 𝑥𝜓
64, 5nfim 1977 . 2 𝑥(𝐹 = (𝑥𝐷𝐵) → 𝜓)
7 fvmptdf.1 . . . 4 (𝜑𝐴𝐷)
87elexd 3366 . . 3 (𝜑𝐴 ∈ V)
9 isset 3359 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
108, 9sylib 208 . 2 (𝜑 → ∃𝑥 𝑥 = 𝐴)
11 fveq1 6331 . . 3 (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴))
12 simpr 471 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐴)
1312fveq2d 6336 . . . . . 6 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝑥) = ((𝑥𝐷𝐵)‘𝐴))
147adantr 466 . . . . . . . 8 ((𝜑𝑥 = 𝐴) → 𝐴𝐷)
1512, 14eqeltrd 2850 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥𝐷)
16 fvmptdf.2 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝐵𝑉)
17 eqid 2771 . . . . . . . 8 (𝑥𝐷𝐵) = (𝑥𝐷𝐵)
1817fvmpt2 6433 . . . . . . 7 ((𝑥𝐷𝐵𝑉) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
1915, 16, 18syl2anc 573 . . . . . 6 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
2013, 19eqtr3d 2807 . . . . 5 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝐴) = 𝐵)
2120eqeq2d 2781 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴) ↔ (𝐹𝐴) = 𝐵))
22 fvmptdf.3 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))
2321, 22sylbid 230 . . 3 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴) → 𝜓))
2411, 23syl5 34 . 2 ((𝜑𝑥 = 𝐴) → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
251, 6, 10, 24exlimdd 2244 1 (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631  ∃wex 1852  Ⅎwnf 1856   ∈ wcel 2145  Ⅎwnfc 2900  Vcvv 3351   ↦ cmpt 4863  ‘cfv 6031 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039 This theorem is referenced by:  fvmptdf  6438
 Copyright terms: Public domain W3C validator