Theorem fsetfocdm 8880

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 8879	. . 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 7110	. . . . 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 7221	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝑔) ∈ V)
9		feq1 6691	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝑔) → (𝑓:𝐴⟶𝐵 ↔ (𝑥 ∈ 𝐴 ↦ 𝑔):𝐴⟶𝐵))
10	9, 1	elab2g 3664	. . . . . 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 6880	. . . . . . . . 9 ⊢ (𝑔 = 𝑓 → (𝑔‘𝑋) = (𝑓‘𝑋))
14	13	cbvmptv 5230	. . . . . . . 8 ⊢ (𝑔 ∈ 𝐹 ↦ (𝑔‘𝑋)) = (𝑓 ∈ 𝐹 ↦ (𝑓‘𝑋))
15	2, 14	eqtri 2759	. . . . . . 7 ⊢ 𝑆 = (𝑓 ∈ 𝐹 ↦ (𝑓‘𝑋))
16		fveq1 6880	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝑔) → (𝑓‘𝑋) = ((𝑥 ∈ 𝐴 ↦ 𝑔)‘𝑋))
17		fvexd 6896	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝑔)‘𝑋) ∈ V)
18	15, 16, 12, 17	fvmptd3 7014	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑆‘(𝑥 ∈ 𝐴 ↦ 𝑔)) = ((𝑥 ∈ 𝐴 ↦ 𝑔)‘𝑋))
19		eqidd 2737	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → 𝑔 = 𝑔)
20		eqid 2736	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑔) = (𝑥 ∈ 𝐴 ↦ 𝑔)
21		vex 3468	. . . . . . . 8 ⊢ 𝑔 ∈ V
22	19, 20, 21	fvmpt 6991	. . . . . . 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 2771	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑆‘(𝑥 ∈ 𝐴 ↦ 𝑔)) = 𝑔)
25		fveq2 6881	. . . . . 6 ⊢ (ℎ = (𝑥 ∈ 𝐴 ↦ 𝑔) → (𝑆‘ℎ) = (𝑆‘(𝑥 ∈ 𝐴 ↦ 𝑔)))
26	25	eqcomd 2742	. . . . 5 ⊢ (ℎ = (𝑥 ∈ 𝐴 ↦ 𝑔) → (𝑆‘(𝑥 ∈ 𝐴 ↦ 𝑔)) = (𝑆‘ℎ))
27	24, 26	sylan9req 2792	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) ∧ ℎ = (𝑥 ∈ 𝐴 ↦ 𝑔)) → 𝑔 = (𝑆‘ℎ))
28	12, 27	rspcedeq2vd 3614	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → ∃ℎ ∈ 𝐹 𝑔 = (𝑆‘ℎ))
29	28	ralrimiva 3133	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ∀𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐹 𝑔 = (𝑆‘ℎ))
30		dffo3 7097	. 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 1540   ∈ wcel 2109  {cab 2714  ∀wral 3052  ∃wrex 3061  Vcvv 3464   ↦ cmpt 5206  ⟶wf 6532  –onto→wfo 6534  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  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 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544

This theorem is referenced by:  fsetprcnex  8881