Theorem fsetfocdm 8859

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 8858	. . 3 ⊢ (𝑋 ∈ 𝐴 → 𝑆:𝐹⟶𝐵)
4	3	adantl 486	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑆:𝐹⟶𝐵)
5		simplr 780	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑔 ∈ 𝐵)
6	5	fmpttd 7112	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝑔):𝐴⟶𝐵)
7		simpll 778	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → 𝐴 ∈ 𝑉)
8	7	mptexd 7224	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝑔) ∈ V)
9		feq1 6685	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝑔) → (𝑓:𝐴⟶𝐵 ↔ (𝑥 ∈ 𝐴 ↦ 𝑔):𝐴⟶𝐵))
10	9, 1	elab2g 3640	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝑔) ∈ V → ((𝑥 ∈ 𝐴 ↦ 𝑔) ∈ 𝐹 ↔ (𝑥 ∈ 𝐴 ↦ 𝑔):𝐴⟶𝐵))
11	8, 10	syl 18	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝑔) ∈ 𝐹 ↔ (𝑥 ∈ 𝐴 ↦ 𝑔):𝐴⟶𝐵))
12	6, 11	mpbird 260	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝑔) ∈ 𝐹)
13		fveq1 6882	. . . . . . . . 9 ⊢ (𝑔 = 𝑓 → (𝑔‘𝑋) = (𝑓‘𝑋))
14	13	cbvmptv 5216	. . . . . . . 8 ⊢ (𝑔 ∈ 𝐹 ↦ (𝑔‘𝑋)) = (𝑓 ∈ 𝐹 ↦ (𝑓‘𝑋))
15	2, 14	eqtri 2786	. . . . . . 7 ⊢ 𝑆 = (𝑓 ∈ 𝐹 ↦ (𝑓‘𝑋))
16		fveq1 6882	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝑔) → (𝑓‘𝑋) = ((𝑥 ∈ 𝐴 ↦ 𝑔)‘𝑋))
17		fvexd 6898	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝑔)‘𝑋) ∈ V)
18	15, 16, 12, 17	fvmptd3 7015	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑆‘(𝑥 ∈ 𝐴 ↦ 𝑔)) = ((𝑥 ∈ 𝐴 ↦ 𝑔)‘𝑋))
19		eqidd 2764	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → 𝑔 = 𝑔)
20		eqid 2763	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑔) = (𝑥 ∈ 𝐴 ↦ 𝑔)
21		vex 3459	. . . . . . . 8 ⊢ 𝑔 ∈ V
22	19, 20, 21	fvmpt 6991	. . . . . . 7 ⊢ (𝑋 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝑔)‘𝑋) = 𝑔)
23	22	ad2antlr 739	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝑔)‘𝑋) = 𝑔)
24	18, 23	eqtrd 2798	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑆‘(𝑥 ∈ 𝐴 ↦ 𝑔)) = 𝑔)
25		fveq2 6883	. . . . . 6 ⊢ (ℎ = (𝑥 ∈ 𝐴 ↦ 𝑔) → (𝑆‘ℎ) = (𝑆‘(𝑥 ∈ 𝐴 ↦ 𝑔)))
26	25	eqcomd 2769	. . . . 5 ⊢ (ℎ = (𝑥 ∈ 𝐴 ↦ 𝑔) → (𝑆‘(𝑥 ∈ 𝐴 ↦ 𝑔)) = (𝑆‘ℎ))
27	24, 26	sylan9req 2819	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) ∧ ℎ = (𝑥 ∈ 𝐴 ↦ 𝑔)) → 𝑔 = (𝑆‘ℎ))
28	12, 27	rspcedeq2vd 3590	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → ∃ℎ ∈ 𝐹 𝑔 = (𝑆‘ℎ))
29	28	ralrimiva 3157	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ∀𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐹 𝑔 = (𝑆‘ℎ))
30		dffo3 7099	. 2 ⊢ (𝑆:𝐹–onto→𝐵 ↔ (𝑆:𝐹⟶𝐵 ∧ ∀𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐹 𝑔 = (𝑆‘ℎ)))
31	4, 29, 30	sylanbrc 594	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑆:𝐹–onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {cab 2741  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ↦ cmpt 5193  ⟶wf 6534  –onto→wfo 6536  ‘cfv 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546

This theorem is referenced by:  fsetprcnex  8860