Theorem fsetsnfo 47773

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 47771	. 2 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵⟶𝐴)
4		vex 3459	. . . . . 6 ⊢ 𝑚 ∈ V
5		eqeq1 2767	. . . . . . 7 ⊢ (𝑦 = 𝑚 → (𝑦 = {⟨𝑆, 𝑏⟩} ↔ 𝑚 = {⟨𝑆, 𝑏⟩}))
6	5	rexbidv 3189	. . . . . 6 ⊢ (𝑦 = 𝑚 → (∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩} ↔ ∃𝑏 ∈ 𝐵 𝑚 = {⟨𝑆, 𝑏⟩}))
7	4, 6, 1	elab2 3642	. . . . 5 ⊢ (𝑚 ∈ 𝐴 ↔ ∃𝑏 ∈ 𝐵 𝑚 = {⟨𝑆, 𝑏⟩})
8		opeq2 4840	. . . . . . . . 9 ⊢ (𝑏 = 𝑛 → ⟨𝑆, 𝑏⟩ = ⟨𝑆, 𝑛⟩)
9	8	sneqd 4602	. . . . . . . 8 ⊢ (𝑏 = 𝑛 → {⟨𝑆, 𝑏⟩} = {⟨𝑆, 𝑛⟩})
10	9	eqeq2d 2774	. . . . . . 7 ⊢ (𝑏 = 𝑛 → (𝑚 = {⟨𝑆, 𝑏⟩} ↔ 𝑚 = {⟨𝑆, 𝑛⟩}))
11	10	cbvrexvw 3244	. . . . . 6 ⊢ (∃𝑏 ∈ 𝐵 𝑚 = {⟨𝑆, 𝑏⟩} ↔ ∃𝑛 ∈ 𝐵 𝑚 = {⟨𝑆, 𝑛⟩})
12		simpr 489	. . . . . . . . 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 4602	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑛⟩})
16	15	adantl 486	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) ∧ 𝑥 = 𝑛) → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑛⟩})
17		simpr 489	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) → 𝑛 ∈ 𝐵)
18		snex 5412	. . . . . . . . . . . . 13 ⊢ {⟨𝑆, 𝑛⟩} ∈ V
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) → {⟨𝑆, 𝑛⟩} ∈ V)
20	13, 16, 17, 19	fvmptd 6999	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) → (𝐹‘𝑛) = {⟨𝑆, 𝑛⟩})
21	20	eqcomd 2769	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) → {⟨𝑆, 𝑛⟩} = (𝐹‘𝑛))
22	21	adantr 485	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) ∧ 𝑚 = {⟨𝑆, 𝑛⟩}) → {⟨𝑆, 𝑛⟩} = (𝐹‘𝑛))
23	12, 22	eqtrd 2798	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) ∧ 𝑚 = {⟨𝑆, 𝑛⟩}) → 𝑚 = (𝐹‘𝑛))
24	23	ex 417	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑛 ∈ 𝐵) → (𝑚 = {⟨𝑆, 𝑛⟩} → 𝑚 = (𝐹‘𝑛)))
25	24	reximdva 3178	. . . . . 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 411	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑚 ∈ 𝐴) → ∃𝑛 ∈ 𝐵 𝑚 = (𝐹‘𝑛))
29	28	ralrimiva 3157	. 2 ⊢ (𝑆 ∈ 𝑉 → ∀𝑚 ∈ 𝐴 ∃𝑛 ∈ 𝐵 𝑚 = (𝐹‘𝑛))
30		dffo3 7099	. 2 ⊢ (𝐹:𝐵–onto→𝐴 ↔ (𝐹:𝐵⟶𝐴 ∧ ∀𝑚 ∈ 𝐴 ∃𝑛 ∈ 𝐵 𝑚 = (𝐹‘𝑛)))
31	3, 29, 30	sylanbrc 594	1 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–onto→𝐴)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {cab 2741  ∀wral 3079  ∃wrex 3089  Vcvv 3455  {csn 4590  ⟨cop 4596   ↦ 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-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-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-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-fo 6544  df-fv 6546

This theorem is referenced by:  fsetsnf1o  47774