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Theorem fmptf 41813
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 1915 . . 3 𝑦 𝐶𝐵
2 nfcsb1v 3879 . . . 4 𝑥𝑦 / 𝑥𝐶
3 fmptf.1 . . . 4 𝑥𝐵
42, 3nfel 2993 . . 3 𝑥𝑦 / 𝑥𝐶𝐵
5 csbeq1a 3869 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
65eleq1d 2898 . . 3 (𝑥 = 𝑦 → (𝐶𝐵𝑦 / 𝑥𝐶𝐵))
71, 4, 6cbvralw 3415 . 2 (∀𝑥𝐴 𝐶𝐵 ↔ ∀𝑦𝐴 𝑦 / 𝑥𝐶𝐵)
8 fmptf.2 . . . 4 𝐹 = (𝑥𝐴𝐶)
9 nfcv 2979 . . . . 5 𝑦𝐶
109, 2, 5cbvmpt 5143 . . . 4 (𝑥𝐴𝐶) = (𝑦𝐴𝑦 / 𝑥𝐶)
118, 10eqtri 2845 . . 3 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐶)
1211fmpt 6856 . 2 (∀𝑦𝐴 𝑦 / 𝑥𝐶𝐵𝐹:𝐴𝐵)
137, 12bitri 278 1 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  wcel 2114  wnfc 2960  wral 3130  csb 3855  cmpt 5122  wf 6330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342
This theorem is referenced by:  rnmptssf  41823
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