Theorem fsetfcdm 8864

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 3457	. . . . 5 ⊢ 𝑔 ∈ V
2		feq1 6684	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓:𝐴⟶𝐵 ↔ 𝑔:𝐴⟶𝐵))
3		fsetfocdm.f	. . . . 5 ⊢ 𝐹 = {𝑓 ∣ 𝑓:𝐴⟶𝐵}
4	1, 2, 3	elab2 3639	. . . 4 ⊢ (𝑔 ∈ 𝐹 ↔ 𝑔:𝐴⟶𝐵)
5		ffvelcdm 7077	. . . . 5 ⊢ ((𝑔:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → (𝑔‘𝑋) ∈ 𝐵)
6	5	expcom 419	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (𝑔:𝐴⟶𝐵 → (𝑔‘𝑋) ∈ 𝐵))
7	4, 6	biimtrid 245	. . 3 ⊢ (𝑋 ∈ 𝐴 → (𝑔 ∈ 𝐹 → (𝑔‘𝑋) ∈ 𝐵))
8	7	imp 412	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑔 ∈ 𝐹) → (𝑔‘𝑋) ∈ 𝐵)
9		fsetfocdm.s	. 2 ⊢ 𝑆 = (𝑔 ∈ 𝐹 ↦ (𝑔‘𝑋))
10	8, 9	fmptd 7110	1 ⊢ (𝑋 ∈ 𝐴 → 𝑆:𝐹⟶𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {cab 2740   ↦ cmpt 5190  ⟶wf 6533  ‘cfv 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545

This theorem is used by:  fsetfocdm  8865