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Theorem fmptf 41385
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 1906 . . 3 𝑦 𝐶𝐵
2 nfcsb1v 3904 . . . 4 𝑥𝑦 / 𝑥𝐶
3 fmptf.1 . . . 4 𝑥𝐵
42, 3nfel 2989 . . 3 𝑥𝑦 / 𝑥𝐶𝐵
5 csbeq1a 3894 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
65eleq1d 2894 . . 3 (𝑥 = 𝑦 → (𝐶𝐵𝑦 / 𝑥𝐶𝐵))
71, 4, 6cbvralw 3439 . 2 (∀𝑥𝐴 𝐶𝐵 ↔ ∀𝑦𝐴 𝑦 / 𝑥𝐶𝐵)
8 fmptf.2 . . . 4 𝐹 = (𝑥𝐴𝐶)
9 nfcv 2974 . . . . 5 𝑦𝐶
109, 2, 5cbvmpt 5158 . . . 4 (𝑥𝐴𝐶) = (𝑦𝐴𝑦 / 𝑥𝐶)
118, 10eqtri 2841 . . 3 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐶)
1211fmpt 6866 . 2 (∀𝑦𝐴 𝑦 / 𝑥𝐶𝐵𝐹:𝐴𝐵)
137, 12bitri 276 1 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  wcel 2105  wnfc 2958  wral 3135  csb 3880  cmpt 5137  wf 6344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356
This theorem is referenced by:  rnmptssf  41395
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