Theorem fsetfocdm 8785

Description: The class of functions with a given domain that is a set and a given codomain is mapped, through evaluation at a point of the domain, onto 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
fsetfocdm	⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑆:𝐹–onto→𝐵)

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

Allowed substitution hint:   𝑆(𝑓)

Proof of Theorem fsetfocdm

Dummy variables ℎ 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsetfocdm.f	. . . 4 ⊢ 𝐹 = {𝑓 ∣ 𝑓:𝐴⟶𝐵}
2		fsetfocdm.s	. . . 4 ⊢ 𝑆 = (𝑔 ∈ 𝐹 ↦ (𝑔‘𝑋))
3	1, 2	fsetfcdm 8784	. . 3 ⊢ (𝑋 ∈ 𝐴 → 𝑆:𝐹⟶𝐵)
4	3	adantl 481	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑆:𝐹⟶𝐵)
5		simplr 768	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑔 ∈ 𝐵)
6	5	fmpttd 7048	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝑔):𝐴⟶𝐵)
7		simpll 766	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → 𝐴 ∈ 𝑉)
8	7	mptexd 7158	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝑔) ∈ V)
9		feq1 6629	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝑔) → (𝑓:𝐴⟶𝐵 ↔ (𝑥 ∈ 𝐴 ↦ 𝑔):𝐴⟶𝐵))
10	9, 1	elab2g 3636	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝑔) ∈ V → ((𝑥 ∈ 𝐴 ↦ 𝑔) ∈ 𝐹 ↔ (𝑥 ∈ 𝐴 ↦ 𝑔):𝐴⟶𝐵))
11	8, 10	syl 17	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝑔) ∈ 𝐹 ↔ (𝑥 ∈ 𝐴 ↦ 𝑔):𝐴⟶𝐵))
12	6, 11	mpbird 257	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝑔) ∈ 𝐹)
13		fveq1 6821	. . . . . . . . 9 ⊢ (𝑔 = 𝑓 → (𝑔‘𝑋) = (𝑓‘𝑋))
14	13	cbvmptv 5195	. . . . . . . 8 ⊢ (𝑔 ∈ 𝐹 ↦ (𝑔‘𝑋)) = (𝑓 ∈ 𝐹 ↦ (𝑓‘𝑋))
15	2, 14	eqtri 2754	. . . . . . 7 ⊢ 𝑆 = (𝑓 ∈ 𝐹 ↦ (𝑓‘𝑋))
16		fveq1 6821	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝑔) → (𝑓‘𝑋) = ((𝑥 ∈ 𝐴 ↦ 𝑔)‘𝑋))
17		fvexd 6837	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝑔)‘𝑋) ∈ V)
18	15, 16, 12, 17	fvmptd3 6952	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑆‘(𝑥 ∈ 𝐴 ↦ 𝑔)) = ((𝑥 ∈ 𝐴 ↦ 𝑔)‘𝑋))
19		eqidd 2732	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → 𝑔 = 𝑔)
20		eqid 2731	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑔) = (𝑥 ∈ 𝐴 ↦ 𝑔)
21		vex 3440	. . . . . . . 8 ⊢ 𝑔 ∈ V
22	19, 20, 21	fvmpt 6929	. . . . . . 7 ⊢ (𝑋 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝑔)‘𝑋) = 𝑔)
23	22	ad2antlr 727	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝑔)‘𝑋) = 𝑔)
24	18, 23	eqtrd 2766	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑆‘(𝑥 ∈ 𝐴 ↦ 𝑔)) = 𝑔)
25		fveq2 6822	. . . . . 6 ⊢ (ℎ = (𝑥 ∈ 𝐴 ↦ 𝑔) → (𝑆‘ℎ) = (𝑆‘(𝑥 ∈ 𝐴 ↦ 𝑔)))
26	25	eqcomd 2737	. . . . 5 ⊢ (ℎ = (𝑥 ∈ 𝐴 ↦ 𝑔) → (𝑆‘(𝑥 ∈ 𝐴 ↦ 𝑔)) = (𝑆‘ℎ))
27	24, 26	sylan9req 2787	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) ∧ ℎ = (𝑥 ∈ 𝐴 ↦ 𝑔)) → 𝑔 = (𝑆‘ℎ))
28	12, 27	rspcedeq2vd 3585	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → ∃ℎ ∈ 𝐹 𝑔 = (𝑆‘ℎ))
29	28	ralrimiva 3124	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ∀𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐹 𝑔 = (𝑆‘ℎ))
30		dffo3 7035	. 2 ⊢ (𝑆:𝐹–onto→𝐵 ↔ (𝑆:𝐹⟶𝐵 ∧ ∀𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐹 𝑔 = (𝑆‘ℎ)))
31	4, 29, 30	sylanbrc 583	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑆:𝐹–onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ↦ cmpt 5172  ⟶wf 6477  –onto→wfo 6479  ‘cfv 6481

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pr 5370

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489

This theorem is referenced by:  fsetprcnex  8786