Theorem fsetsnfo 47070

Description: The mapping of an element of a class to a singleton function is a surjection. (Contributed by AV, 13-Sep-2024.)

Hypotheses

Ref	Expression
fsetsnf.a	⊢ 𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}
fsetsnf.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})

Assertion

Ref	Expression
fsetsnfo	⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–onto→𝐴)

Distinct variable groups:   𝑥,𝐴   𝐵,𝑏,𝑥,𝑦   𝑆,𝑏,𝑥,𝑦   𝑉,𝑏,𝑥

Allowed substitution hints:   𝐴(𝑦,𝑏)   𝐹(𝑥,𝑦,𝑏)   𝑉(𝑦)

Proof of Theorem fsetsnfo

Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsetsnf.a	. . 3 ⊢ 𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}
2		fsetsnf.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})
3	1, 2	fsetsnf 47068	. 2 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵⟶𝐴)
4		vex 3483	. . . . . 6 ⊢ 𝑚 ∈ V
5		eqeq1 2740	. . . . . . 7 ⊢ (𝑦 = 𝑚 → (𝑦 = {⟨𝑆, 𝑏⟩} ↔ 𝑚 = {⟨𝑆, 𝑏⟩}))
6	5	rexbidv 3178	. . . . . 6 ⊢ (𝑦 = 𝑚 → (∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩} ↔ ∃𝑏 ∈ 𝐵 𝑚 = {⟨𝑆, 𝑏⟩}))
7	4, 6, 1	elab2 3681	. . . . 5 ⊢ (𝑚 ∈ 𝐴 ↔ ∃𝑏 ∈ 𝐵 𝑚 = {⟨𝑆, 𝑏⟩})
8		opeq2 4873	. . . . . . . . 9 ⊢ (𝑏 = 𝑛 → ⟨𝑆, 𝑏⟩ = ⟨𝑆, 𝑛⟩)
9	8	sneqd 4637	. . . . . . . 8 ⊢ (𝑏 = 𝑛 → {⟨𝑆, 𝑏⟩} = {⟨𝑆, 𝑛⟩})
10	9	eqeq2d 2747	. . . . . . 7 ⊢ (𝑏 = 𝑛 → (𝑚 = {⟨𝑆, 𝑏⟩} ↔ 𝑚 = {⟨𝑆, 𝑛⟩}))
11	10	cbvrexvw 3237	. . . . . 6 ⊢ (∃𝑏 ∈ 𝐵 𝑚 = {⟨𝑆, 𝑏⟩} ↔ ∃𝑛 ∈ 𝐵 𝑚 = {⟨𝑆, 𝑛⟩})
12		simpr 484	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) ∧ 𝑚 = {⟨𝑆, 𝑛⟩}) → 𝑚 = {⟨𝑆, 𝑛⟩})
13	2	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) → 𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩}))
14		opeq2 4873	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑛 → ⟨𝑆, 𝑥⟩ = ⟨𝑆, 𝑛⟩)
15	14	sneqd 4637	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑛⟩})
16	15	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) ∧ 𝑥 = 𝑛) → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑛⟩})
17		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) → 𝑛 ∈ 𝐵)
18		snex 5435	. . . . . . . . . . . . 13 ⊢ {⟨𝑆, 𝑛⟩} ∈ V
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) → {⟨𝑆, 𝑛⟩} ∈ V)
20	13, 16, 17, 19	fvmptd 7022	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) → (𝐹‘𝑛) = {⟨𝑆, 𝑛⟩})
21	20	eqcomd 2742	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) → {⟨𝑆, 𝑛⟩} = (𝐹‘𝑛))
22	21	adantr 480	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) ∧ 𝑚 = {⟨𝑆, 𝑛⟩}) → {⟨𝑆, 𝑛⟩} = (𝐹‘𝑛))
23	12, 22	eqtrd 2776	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) ∧ 𝑚 = {⟨𝑆, 𝑛⟩}) → 𝑚 = (𝐹‘𝑛))
24	23	ex 412	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) → (𝑚 = {⟨𝑆, 𝑛⟩} → 𝑚 = (𝐹‘𝑛)))
25	24	reximdva 3167	. . . . . 6 ⊢ (𝑆 ∈ 𝑉 → (∃𝑛 ∈ 𝐵 𝑚 = {⟨𝑆, 𝑛⟩} → ∃𝑛 ∈ 𝐵 𝑚 = (𝐹‘𝑛)))
26	11, 25	biimtrid 242	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (∃𝑏 ∈ 𝐵 𝑚 = {⟨𝑆, 𝑏⟩} → ∃𝑛 ∈ 𝐵 𝑚 = (𝐹‘𝑛)))
27	7, 26	biimtrid 242	. . . 4 ⊢ (𝑆 ∈ 𝑉 → (𝑚 ∈ 𝐴 → ∃𝑛 ∈ 𝐵 𝑚 = (𝐹‘𝑛)))
28	27	imp 406	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑚 ∈ 𝐴) → ∃𝑛 ∈ 𝐵 𝑚 = (𝐹‘𝑛))
29	28	ralrimiva 3145	. 2 ⊢ (𝑆 ∈ 𝑉 → ∀𝑚 ∈ 𝐴 ∃𝑛 ∈ 𝐵 𝑚 = (𝐹‘𝑛))
30		dffo3 7121	. 2 ⊢ (𝐹:𝐵–onto→𝐴 ↔ (𝐹:𝐵⟶𝐴 ∧ ∀𝑚 ∈ 𝐴 ∃𝑛 ∈ 𝐵 𝑚 = (𝐹‘𝑛)))
31	3, 29, 30	sylanbrc 583	1 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–onto→𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {cab 2713  ∀wral 3060  ∃wrex 3069  Vcvv 3479  {csn 4625  ⟨cop 4631   ↦ cmpt 5224  ⟶wf 6556  –onto→wfo 6558  ‘cfv 6560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fo 6566  df-fv 6568

This theorem is referenced by:  fsetsnf1o  47071