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Theorem fmptd2f 39960
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 2771 . 2 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
41, 2, 3fmptdf 6529 1 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wnf 1856  wcel 2145  cmpt 4863  wf 6027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039
This theorem is referenced by:  fmptd2  39978  climinf2mpt  40464  climinfmpt  40465  limsupvaluzmpt  40467  limsupre2mpt  40480  limsupre3mpt  40484  limsupreuzmpt  40489  supcnvlimsupmpt  40491  liminfvalxrmpt  40536  smflimsupmpt  41555  smfliminfmpt  41558
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