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Theorem fmptd2f 45245
Description: Domain and codomain of the mapping operation; deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
fmptd2f.1 𝑥𝜑
fmptd2f.2 ((𝜑𝑥𝐴) → 𝐵𝐶)
Assertion
Ref Expression
fmptd2f (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem fmptd2f
StepHypRef Expression
1 fmptd2f.1 . 2 𝑥𝜑
2 fmptd2f.2 . 2 ((𝜑𝑥𝐴) → 𝐵𝐶)
3 eqid 2736 . 2 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
41, 2, 3fmptdf 7136 1 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wnf 1782  wcel 2107  cmpt 5224  wf 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-fun 6562  df-fn 6563  df-f 6564
This theorem is referenced by:  climinf2mpt  45734  climinfmpt  45735  limsupvaluzmpt  45737  limsupre2mpt  45750  limsupre3mpt  45754  limsupreuzmpt  45759  supcnvlimsupmpt  45761  liminfvalxrmpt  45806  liminflbuz2  45835  dvnprodlem1  45966  sge0z  46395  sge0f1o  46402  smfsupmpt  46835  smfinfmpt  46839  smflimsupmpt  46849  smfliminfmpt  46852  smfsupdmmbllem  46864  smfinfdmmbllem  46868
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