Theorem fsetfcdm 8834

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 3474	. . . . 5 ⊢ 𝑔 ∈ V
2		feq1 6682	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓:𝐴⟶𝐵 ↔ 𝑔:𝐴⟶𝐵))
3		fsetfocdm.f	. . . . 5 ⊢ 𝐹 = {𝑓 ∣ 𝑓:𝐴⟶𝐵}
4	1, 2, 3	elab2 3665	. . . 4 ⊢ (𝑔 ∈ 𝐹 ↔ 𝑔:𝐴⟶𝐵)
5		ffvelcdm 7065	. . . . 5 ⊢ ((𝑔:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → (𝑔‘𝑋) ∈ 𝐵)
6	5	expcom 414	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (𝑔:𝐴⟶𝐵 → (𝑔‘𝑋) ∈ 𝐵))
7	4, 6	biimtrid 241	. . 3 ⊢ (𝑋 ∈ 𝐴 → (𝑔 ∈ 𝐹 → (𝑔‘𝑋) ∈ 𝐵))
8	7	imp 407	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑔 ∈ 𝐹) → (𝑔‘𝑋) ∈ 𝐵)
9		fsetfocdm.s	. 2 ⊢ 𝑆 = (𝑔 ∈ 𝐹 ↦ (𝑔‘𝑋))
10	8, 9	fmptd 7095	1 ⊢ (𝑋 ∈ 𝐴 → 𝑆:𝐹⟶𝐵)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  {cab 2708   ↦ cmpt 5221  ⟶wf 6525  ‘cfv 6529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3430  df-v 3472  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4520  df-sn 4620  df-pr 4622  df-op 4626  df-uni 4899  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6481  df-fun 6531  df-fn 6532  df-f 6533  df-fv 6537

This theorem is referenced by:  fsetfocdm  8835