Theorem fvmptelcdm 7054

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 2739	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	fmpt 7051	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)
4	1, 3	sylibr 235	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
5	4	r19.21bi 3231	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2119  ∀wral 3053   ↦ cmpt 5153  ⟶wf 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-fun 6487  df-fn 6488  df-f 6489

This theorem is referenced by:  rlimmptrcl  15561  lo1mptrcl  15575  o1mptrcl  15576  frlmgsum  21747  uvcresum  21768  psrass1lem  21908  txcnp  23603  ptcnp  23605  ptcn  23610  cnmpt11  23646  cnmpt1t  23648  cnmpt12  23650  cnmptkp  23663  cnmptk1  23664  cnmptkk  23666  cnmptk1p  23668  cnmptk2  23669  cnmpt1plusg  24070  cnmpt1vsca  24177  cnmpt1ds  24826  cncfcompt2  24893  cncfmpt2ss  24901  cnmpt1ip  25232  divcncf  25432  mbfmptcl  25621  i1fposd  25692  itgss3  25800  dvmptcl  25944  dvmptco  25957  dvle  25992  dvfsumle  26006  dvfsumge  26007  dvmptrecl  26009  itgparts  26032  itgsubstlem  26033  itgsubst  26034  ulmss  26380  ulmdvlem2  26384  itgulm2  26392  logtayl  26642  intlewftc  42546  cncfcompt  46326  cncficcgt0  46331  itgsubsticclem  46418  sge0iunmptlemre  46858  hoicvrrex  46999  smfadd  47208  smfpimioompt  47229