Theorem fsetfocdm 8607

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 8606	. . 3 ⊢ (𝑋 ∈ 𝐴 → 𝑆:𝐹⟶𝐵)
4	3	adantl 481	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑆:𝐹⟶𝐵)
5		simplr 765	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑔 ∈ 𝐵)
6		eqid 2738	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑔) = (𝑥 ∈ 𝐴 ↦ 𝑔)
7	5, 6	fmptd 6970	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝑔):𝐴⟶𝐵)
8		simpll 763	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → 𝐴 ∈ 𝑉)
9	8	mptexd 7082	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝑔) ∈ V)
10		feq1 6565	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝑔) → (𝑓:𝐴⟶𝐵 ↔ (𝑥 ∈ 𝐴 ↦ 𝑔):𝐴⟶𝐵))
11	10, 1	elab2g 3604	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝑔) ∈ V → ((𝑥 ∈ 𝐴 ↦ 𝑔) ∈ 𝐹 ↔ (𝑥 ∈ 𝐴 ↦ 𝑔):𝐴⟶𝐵))
12	9, 11	syl 17	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝑔) ∈ 𝐹 ↔ (𝑥 ∈ 𝐴 ↦ 𝑔):𝐴⟶𝐵))
13	7, 12	mpbird 256	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝑔) ∈ 𝐹)
14		fveq2 6756	. . . . . 6 ⊢ (ℎ = (𝑥 ∈ 𝐴 ↦ 𝑔) → (𝑆‘ℎ) = (𝑆‘(𝑥 ∈ 𝐴 ↦ 𝑔)))
15	14	eqeq2d 2749	. . . . 5 ⊢ (ℎ = (𝑥 ∈ 𝐴 ↦ 𝑔) → (𝑔 = (𝑆‘ℎ) ↔ 𝑔 = (𝑆‘(𝑥 ∈ 𝐴 ↦ 𝑔))))
16	15	adantl 481	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) ∧ ℎ = (𝑥 ∈ 𝐴 ↦ 𝑔)) → (𝑔 = (𝑆‘ℎ) ↔ 𝑔 = (𝑆‘(𝑥 ∈ 𝐴 ↦ 𝑔))))
17		fveq1 6755	. . . . . . . . 9 ⊢ (𝑔 = 𝑓 → (𝑔‘𝑋) = (𝑓‘𝑋))
18	17	cbvmptv 5183	. . . . . . . 8 ⊢ (𝑔 ∈ 𝐹 ↦ (𝑔‘𝑋)) = (𝑓 ∈ 𝐹 ↦ (𝑓‘𝑋))
19	2, 18	eqtri 2766	. . . . . . 7 ⊢ 𝑆 = (𝑓 ∈ 𝐹 ↦ (𝑓‘𝑋))
20	19	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → 𝑆 = (𝑓 ∈ 𝐹 ↦ (𝑓‘𝑋)))
21		fveq1 6755	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝑔) → (𝑓‘𝑋) = ((𝑥 ∈ 𝐴 ↦ 𝑔)‘𝑋))
22	21	adantl 481	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝑔)) → (𝑓‘𝑋) = ((𝑥 ∈ 𝐴 ↦ 𝑔)‘𝑋))
23		fvexd 6771	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝑔)‘𝑋) ∈ V)
24	20, 22, 13, 23	fvmptd 6864	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑆‘(𝑥 ∈ 𝐴 ↦ 𝑔)) = ((𝑥 ∈ 𝐴 ↦ 𝑔)‘𝑋))
25		eqidd 2739	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ 𝐴 ↦ 𝑔) = (𝑥 ∈ 𝐴 ↦ 𝑔))
26		eqidd 2739	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 = 𝑋) → 𝑔 = 𝑔)
27		simpr 484	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴)
28		vex 3426	. . . . . . . 8 ⊢ 𝑔 ∈ V
29	28	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑔 ∈ V)
30	25, 26, 27, 29	fvmptd 6864	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝑔)‘𝑋) = 𝑔)
31	30	adantr 480	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝑔)‘𝑋) = 𝑔)
32	24, 31	eqtr2d 2779	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → 𝑔 = (𝑆‘(𝑥 ∈ 𝐴 ↦ 𝑔)))
33	13, 16, 32	rspcedvd 3555	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → ∃ℎ ∈ 𝐹 𝑔 = (𝑆‘ℎ))
34	33	ralrimiva 3107	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ∀𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐹 𝑔 = (𝑆‘ℎ))
35		dffo3 6960	. 2 ⊢ (𝑆:𝐹–onto→𝐵 ↔ (𝑆:𝐹⟶𝐵 ∧ ∀𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐹 𝑔 = (𝑆‘ℎ)))
36	4, 34, 35	sylanbrc 582	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑆:𝐹–onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ↦ cmpt 5153  ⟶wf 6414  –onto→wfo 6416  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426

This theorem is referenced by:  fsetprcnex  8608