Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fmptf Structured version   Visualization version   GIF version

Theorem fmptf 44614
Description: Functionality of the mapping operation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
fmptf.1 𝑥𝐵
fmptf.2 𝐹 = (𝑥𝐴𝐶)
Assertion
Ref Expression
fmptf (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)

Proof of Theorem fmptf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfv 1910 . . 3 𝑦 𝐶𝐵
2 nfcsb1v 3917 . . . 4 𝑥𝑦 / 𝑥𝐶
3 fmptf.1 . . . 4 𝑥𝐵
42, 3nfel 2914 . . 3 𝑥𝑦 / 𝑥𝐶𝐵
5 csbeq1a 3906 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
65eleq1d 2814 . . 3 (𝑥 = 𝑦 → (𝐶𝐵𝑦 / 𝑥𝐶𝐵))
71, 4, 6cbvralw 3300 . 2 (∀𝑥𝐴 𝐶𝐵 ↔ ∀𝑦𝐴 𝑦 / 𝑥𝐶𝐵)
8 fmptf.2 . . . 4 𝐹 = (𝑥𝐴𝐶)
9 nfcv 2899 . . . . 5 𝑦𝐶
109, 2, 5cbvmpt 5259 . . . 4 (𝑥𝐴𝐶) = (𝑦𝐴𝑦 / 𝑥𝐶)
118, 10eqtri 2756 . . 3 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐶)
1211fmpt 7120 . 2 (∀𝑦𝐴 𝑦 / 𝑥𝐶𝐵𝐹:𝐴𝐵)
137, 12bitri 275 1 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1534  wcel 2099  wnfc 2879  wral 3058  csb 3892  cmpt 5231  wf 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-fun 6550  df-fn 6551  df-f 6552
This theorem is referenced by:  rnmptssf  44623
  Copyright terms: Public domain W3C validator