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Theorem fvmptelcdm 7133
Description: The value of a function at a point of its domain belongs to its codomain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypothesis
Ref Expression
fvmptelcdm.1 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
Assertion
Ref Expression
fvmptelcdm ((𝜑𝑥𝐴) → 𝐵𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem fvmptelcdm
StepHypRef Expression
1 fvmptelcdm.1 . . 3 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
2 eqid 2737 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32fmpt 7130 . . 3 (∀𝑥𝐴 𝐵𝐶 ↔ (𝑥𝐴𝐵):𝐴𝐶)
41, 3sylibr 234 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
54r19.21bi 3251 1 ((𝜑𝑥𝐴) → 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  wral 3061  cmpt 5225  wf 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565
This theorem is referenced by:  rlimmptrcl  15644  lo1mptrcl  15658  o1mptrcl  15659  frlmgsum  21792  uvcresum  21813  psrass1lem  21952  txcnp  23628  ptcnp  23630  ptcn  23635  cnmpt11  23671  cnmpt1t  23673  cnmpt12  23675  cnmptkp  23688  cnmptk1  23689  cnmptkk  23691  cnmptk1p  23693  cnmptk2  23694  cnmpt1plusg  24095  cnmpt1vsca  24202  cnmpt1ds  24864  cncfcompt2  24934  cncfmpt2ss  24942  cnmpt1ip  25281  divcncf  25482  mbfmptcl  25671  i1fposd  25742  itgss3  25850  dvmptcl  25997  dvmptco  26010  dvle  26046  dvfsumle  26060  dvfsumleOLD  26061  dvfsumge  26062  dvmptrecl  26064  itgparts  26088  itgsubstlem  26089  itgsubst  26090  ulmss  26440  ulmdvlem2  26444  itgulm2  26452  logtayl  26702  intlewftc  42062  cncfcompt  45898  cncficcgt0  45903  itgsubsticclem  45990  sge0iunmptlemre  46430  hoicvrrex  46571  smfadd  46780  smfpimioompt  46801
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