Theorem fvmptelcdm 7118

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 2760	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	fmpt 7115	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)
4	1, 3	sylibr 237	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
5	4	r19.21bi 3254	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2144  ∀wral 3076   ↦ cmpt 5185  ⟶wf 6539

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-fun 6545  df-fn 6546  df-f 6547

This theorem is used by:  rlimmptrcl  15784  lo1mptrcl  15798  o1mptrcl  15799  frlmgsum  22087  uvcresum  22108  psrass1lem  22250  txcnp  23948  ptcnp  23950  ptcn  23955  cnmpt11  23991  cnmpt1t  23993  cnmpt12  23995  cnmptkp  24008  cnmptk1  24009  cnmptkk  24011  cnmptk1p  24013  cnmptk2  24014  cnmpt1plusg  24415  cnmpt1vsca  24522  cnmpt1ds  25171  cncfcompt2  25238  cncfmpt2ss  25246  cnmpt1ip  25577  divcncf  25777  mbfmptcl  25966  i1fposd  26037  itgss3  26144  dvmptcl  26288  dvmptco  26301  dvle  26336  dvfsumle  26350  dvfsumge  26351  dvmptrecl  26353  itgparts  26376  itgsubstlem  26377  itgsubst  26378  ulmss  26733  ulmdvlem2  26737  itgulm2  26745  logtayl  26997  intlewftc  43220  cncfcompt  47001  cncficcgt0  47006  itgsubsticclem  47093  sge0iunmptlemre  47533  hoicvrrex  47674  smfadd  47883  smfpimioompt  47904