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Theorem fmptf 44487
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 1909 . . 3 𝑦 𝐶𝐵
2 nfcsb1v 3911 . . . 4 𝑥𝑦 / 𝑥𝐶
3 fmptf.1 . . . 4 𝑥𝐵
42, 3nfel 2909 . . 3 𝑥𝑦 / 𝑥𝐶𝐵
5 csbeq1a 3900 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
65eleq1d 2810 . . 3 (𝑥 = 𝑦 → (𝐶𝐵𝑦 / 𝑥𝐶𝐵))
71, 4, 6cbvralw 3295 . 2 (∀𝑥𝐴 𝐶𝐵 ↔ ∀𝑦𝐴 𝑦 / 𝑥𝐶𝐵)
8 fmptf.2 . . . 4 𝐹 = (𝑥𝐴𝐶)
9 nfcv 2895 . . . . 5 𝑦𝐶
109, 2, 5cbvmpt 5250 . . . 4 (𝑥𝐴𝐶) = (𝑦𝐴𝑦 / 𝑥𝐶)
118, 10eqtri 2752 . . 3 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐶)
1211fmpt 7102 . 2 (∀𝑦𝐴 𝑦 / 𝑥𝐶𝐵𝐹:𝐴𝐵)
137, 12bitri 275 1 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wcel 2098  wnfc 2875  wral 3053  csb 3886  cmpt 5222  wf 6530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-fun 6536  df-fn 6537  df-f 6538
This theorem is referenced by:  rnmptssf  44496
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