Theorem fvmptelcdm 7132

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 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)

Assertion

Ref	Expression
fvmptelcdm	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem fvmptelcdm

Step	Hyp	Ref	Expression
1		fvmptelcdm.1	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)
2		eqid 2734	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	fmpt 7129	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)
4	1, 3	sylibr 234	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
5	4	r19.21bi 3248	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2105  ∀wral 3058   ↦ cmpt 5230  ⟶wf 6558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-fun 6564  df-fn 6565  df-f 6566

This theorem is referenced by:  rlimmptrcl  15640  lo1mptrcl  15654  o1mptrcl  15655  frlmgsum  21809  uvcresum  21830  psrass1lem  21969  txcnp  23643  ptcnp  23645  ptcn  23650  cnmpt11  23686  cnmpt1t  23688  cnmpt12  23690  cnmptkp  23703  cnmptk1  23704  cnmptkk  23706  cnmptk1p  23708  cnmptk2  23709  cnmpt1plusg  24110  cnmpt1vsca  24217  cnmpt1ds  24877  cncfcompt2  24947  cncfmpt2ss  24955  cnmpt1ip  25294  divcncf  25495  mbfmptcl  25684  i1fposd  25756  itgss3  25864  dvmptcl  26011  dvmptco  26024  dvle  26060  dvfsumle  26074  dvfsumleOLD  26075  dvfsumge  26076  dvmptrecl  26078  itgparts  26102  itgsubstlem  26103  itgsubst  26104  ulmss  26454  ulmdvlem2  26458  itgulm2  26466  logtayl  26716  intlewftc  42042  cncfcompt  45838  cncficcgt0  45843  itgsubsticclem  45930  sge0iunmptlemre  46370  hoicvrrex  46511  smfadd  46720  smfpimioompt  46741