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Theorem fvmptelrn 6609
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
fvmptelrn.1 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
Assertion
Ref Expression
fvmptelrn ((𝜑𝑥𝐴) → 𝐵𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem fvmptelrn
StepHypRef Expression
1 fvmptelrn.1 . . 3 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
2 eqid 2799 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32fmpt 6606 . . 3 (∀𝑥𝐴 𝐵𝐶 ↔ (𝑥𝐴𝐵):𝐴𝐶)
41, 3sylibr 226 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
54r19.21bi 3113 1 ((𝜑𝑥𝐴) → 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wcel 2157  wral 3089  cmpt 4922  wf 6097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109
This theorem is referenced by:  divcncf  23555  itgulm2  24504  cncfcompt  40840  cncficcgt0  40845  cncfcompt2  40856  itgsubsticclem  40934  sge0iunmptlemre  41375  hoicvrrex  41516  smfadd  41719  smfpimioompt  41739  smfsupmpt  41767  smfinfmpt  41771
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