Theorem fsetfcdm 8795

Description: The class of functions with a given domain and a given codomain is mapped, through evaluation at a point of the domain, into the codomain. (Contributed by AV, 15-Sep-2024.)

Hypotheses

Ref	Expression
fsetfocdm.f	⊢ 𝐹 = {𝑓 ∣ 𝑓:𝐴⟶𝐵}
fsetfocdm.s	⊢ 𝑆 = (𝑔 ∈ 𝐹 ↦ (𝑔‘𝑋))

Assertion

Ref	Expression
fsetfcdm	⊢ (𝑋 ∈ 𝐴 → 𝑆:𝐹⟶𝐵)

Distinct variable groups:   𝐴,𝑓,𝑔   𝐵,𝑓,𝑔   𝑔,𝐹   𝑔,𝑋

Allowed substitution hints:   𝑆(𝑓,𝑔)   𝐹(𝑓)   𝑋(𝑓)

Proof of Theorem fsetfcdm

Step	Hyp	Ref	Expression
1		vex 3442	. . . . 5 ⊢ 𝑔 ∈ V
2		feq1 6638	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓:𝐴⟶𝐵 ↔ 𝑔:𝐴⟶𝐵))
3		fsetfocdm.f	. . . . 5 ⊢ 𝐹 = {𝑓 ∣ 𝑓:𝐴⟶𝐵}
4	1, 2, 3	elab2 3635	. . . 4 ⊢ (𝑔 ∈ 𝐹 ↔ 𝑔:𝐴⟶𝐵)
5		ffvelcdm 7024	. . . . 5 ⊢ ((𝑔:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → (𝑔‘𝑋) ∈ 𝐵)
6	5	expcom 413	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (𝑔:𝐴⟶𝐵 → (𝑔‘𝑋) ∈ 𝐵))
7	4, 6	biimtrid 242	. . 3 ⊢ (𝑋 ∈ 𝐴 → (𝑔 ∈ 𝐹 → (𝑔‘𝑋) ∈ 𝐵))
8	7	imp 406	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑔 ∈ 𝐹) → (𝑔‘𝑋) ∈ 𝐵)
9		fsetfocdm.s	. 2 ⊢ 𝑆 = (𝑔 ∈ 𝐹 ↦ (𝑔‘𝑋))
10	8, 9	fmptd 7057	1 ⊢ (𝑋 ∈ 𝐴 → 𝑆:𝐹⟶𝐵)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  {cab 2712   ↦ cmpt 5177  ⟶wf 6486  ‘cfv 6490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498

This theorem is referenced by:  fsetfocdm  8796