Theorem fsetsnfo 48216

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 48214	. 2 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵⟶𝐴)
4		vex 3454	. . . . . 6 ⊢ 𝑚 ∈ V
5		eqeq1 2764	. . . . . . 7 ⊢ (𝑦 = 𝑚 → (𝑦 = {⟨𝑆, 𝑏⟩} ↔ 𝑚 = {⟨𝑆, 𝑏⟩}))
6	5	rexbidv 3186	. . . . . 6 ⊢ (𝑦 = 𝑚 → (∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩} ↔ ∃𝑏 ∈ 𝐵 𝑚 = {⟨𝑆, 𝑏⟩}))
7	4, 6, 1	elab2 3635	. . . . 5 ⊢ (𝑚 ∈ 𝐴 ↔ ∃𝑏 ∈ 𝐵 𝑚 = {⟨𝑆, 𝑏⟩})
8		opeq2 4840	. . . . . . . . 9 ⊢ (𝑏 = 𝑛 → ⟨𝑆, 𝑏⟩ = ⟨𝑆, 𝑛⟩)
9	8	sneqd 4592	. . . . . . . 8 ⊢ (𝑏 = 𝑛 → {⟨𝑆, 𝑏⟩} = {⟨𝑆, 𝑛⟩})
10	9	eqeq2d 2771	. . . . . . 7 ⊢ (𝑏 = 𝑛 → (𝑚 = {⟨𝑆, 𝑏⟩} ↔ 𝑚 = {⟨𝑆, 𝑛⟩}))
11	10	cbvrexvw 3241	. . . . . 6 ⊢ (∃𝑏 ∈ 𝐵 𝑚 = {⟨𝑆, 𝑏⟩} ↔ ∃𝑛 ∈ 𝐵 𝑚 = {⟨𝑆, 𝑛⟩})
12		simpr 490	. . . . . . . . 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 4840	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑛 → ⟨𝑆, 𝑥⟩ = ⟨𝑆, 𝑛⟩)
15	14	sneqd 4592	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑛⟩})
16	15	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) ∧ 𝑥 = 𝑛) → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑛⟩})
17		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) → 𝑛 ∈ 𝐵)
18		snex 5396	. . . . . . . . . . . . 13 ⊢ {⟨𝑆, 𝑛⟩} ∈ V
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) → {⟨𝑆, 𝑛⟩} ∈ V)
20	13, 16, 17, 19	fvmptd 7001	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) → (𝐹‘𝑛) = {⟨𝑆, 𝑛⟩})
21	20	eqcomd 2766	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) → {⟨𝑆, 𝑛⟩} = (𝐹‘𝑛))
22	21	adantr 486	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) ∧ 𝑚 = {⟨𝑆, 𝑛⟩}) → {⟨𝑆, 𝑛⟩} = (𝐹‘𝑛))
23	12, 22	eqtrd 2795	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) ∧ 𝑚 = {⟨𝑆, 𝑛⟩}) → 𝑚 = (𝐹‘𝑛))
24	23	ex 418	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) → (𝑚 = {⟨𝑆, 𝑛⟩} → 𝑚 = (𝐹‘𝑛)))
25	24	reximdva 3175	. . . . . 6 ⊢ (𝑆 ∈ 𝑉 → (∃𝑛 ∈ 𝐵 𝑚 = {⟨𝑆, 𝑛⟩} → ∃𝑛 ∈ 𝐵 𝑚 = (𝐹‘𝑛)))
26	11, 25	biimtrid 245	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (∃𝑏 ∈ 𝐵 𝑚 = {⟨𝑆, 𝑏⟩} → ∃𝑛 ∈ 𝐵 𝑚 = (𝐹‘𝑛)))
27	7, 26	biimtrid 245	. . . 4 ⊢ (𝑆 ∈ 𝑉 → (𝑚 ∈ 𝐴 → ∃𝑛 ∈ 𝐵 𝑚 = (𝐹‘𝑛)))
28	27	imp 412	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑚 ∈ 𝐴) → ∃𝑛 ∈ 𝐵 𝑚 = (𝐹‘𝑛))
29	28	ralrimiva 3154	. 2 ⊢ (𝑆 ∈ 𝑉 → ∀𝑚 ∈ 𝐴 ∃𝑛 ∈ 𝐵 𝑚 = (𝐹‘𝑛))
30		dffo3 7102	. 2 ⊢ (𝐹:𝐵–onto→𝐴 ↔ (𝐹:𝐵⟶𝐴 ∧ ∀𝑚 ∈ 𝐴 ∃𝑛 ∈ 𝐵 𝑚 = (𝐹‘𝑛)))
31	3, 29, 30	sylanbrc 595	1 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–onto→𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086  Vcvv 3450  {csn 4584  ⟨cop 4834   ↦ cmpt 5185  ⟶wf 6534  –onto→wfo 6536  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546

This theorem is used by:  fsetsnf1o  48217