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Theorem fmptd2f 39756
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 2651 . 2 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
41, 2, 3fmptdf 6427 1 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wnf 1748  wcel 2030  cmpt 4762  wf 5922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934
This theorem is referenced by:  fmptd2  39774  climinf2mpt  40264  climinfmpt  40265  limsupvaluzmpt  40267  limsupre2mpt  40280  limsupre3mpt  40284  limsupreuzmpt  40289  supcnvlimsupmpt  40291  liminfvalxrmpt  40336  smflimsupmpt  41356  smfliminfmpt  41359
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