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Theorem fmptd2 39975
 Description: Domain and codomain of the mapping operation; deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypothesis
Ref Expression
fmptd2.1 ((𝜑𝑥𝐴) → 𝐵𝐶)
Assertion
Ref Expression
fmptd2 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝜑,𝑥
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem fmptd2
StepHypRef Expression
1 nfv 1995 . 2 𝑥𝜑
2 fmptd2.1 . 2 ((𝜑𝑥𝐴) → 𝐵𝐶)
31, 2fmptd2f 39957 1 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   ∈ wcel 2145   ↦ cmpt 4864  ⟶wf 6026 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 4916  ax-nul 4924  ax-pr 5035 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 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038 This theorem is referenced by:  liminfltlem  40551
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