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Theorem fmptf 39966
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 1995 . . 3 𝑦 𝐶𝐵
2 nfcsb1v 3698 . . . 4 𝑥𝑦 / 𝑥𝐶
3 fmptf.1 . . . 4 𝑥𝐵
42, 3nfel 2926 . . 3 𝑥𝑦 / 𝑥𝐶𝐵
5 csbeq1a 3691 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
65eleq1d 2835 . . 3 (𝑥 = 𝑦 → (𝐶𝐵𝑦 / 𝑥𝐶𝐵))
71, 4, 6cbvral 3316 . 2 (∀𝑥𝐴 𝐶𝐵 ↔ ∀𝑦𝐴 𝑦 / 𝑥𝐶𝐵)
8 fmptf.2 . . . 4 𝐹 = (𝑥𝐴𝐶)
9 nfcv 2913 . . . . 5 𝑦𝐶
109, 2, 5cbvmpt 4883 . . . 4 (𝑥𝐴𝐶) = (𝑦𝐴𝑦 / 𝑥𝐶)
118, 10eqtri 2793 . . 3 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐶)
1211fmpt 6523 . 2 (∀𝑦𝐴 𝑦 / 𝑥𝐶𝐵𝐹:𝐴𝐵)
137, 12bitri 264 1 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1631  wcel 2145  wnfc 2900  wral 3061  csb 3682  cmpt 4863  wf 6027
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-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  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-ne 2944  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-fn 6034  df-f 6035  df-fv 6039
This theorem is referenced by:  rnmptssf  39980
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