Theorem fsetsnf1 45841

Description: The mapping of an element of a class to a singleton function is an injection. (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
fsetsnf1	⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–1-1→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 fsetsnf1

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 45840	. 2 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵⟶𝐴)
4	2	a1i 11	. . . . . . 7 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → 𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩}))
5		opeq2 4874	. . . . . . . . 9 ⊢ (𝑥 = 𝑚 → ⟨𝑆, 𝑥⟩ = ⟨𝑆, 𝑚⟩)
6	5	sneqd 4640	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑚⟩})
7	6	adantl 482	. . . . . . 7 ⊢ (((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) ∧ 𝑥 = 𝑚) → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑚⟩})
8		simpl 483	. . . . . . 7 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → 𝑚 ∈ 𝐵)
9		snex 5431	. . . . . . . 8 ⊢ {⟨𝑆, 𝑚⟩} ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → {⟨𝑆, 𝑚⟩} ∈ V)
11	4, 7, 8, 10	fvmptd 7005	. . . . . 6 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → (𝐹‘𝑚) = {⟨𝑆, 𝑚⟩})
12		opeq2 4874	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → ⟨𝑆, 𝑥⟩ = ⟨𝑆, 𝑛⟩)
13	12	sneqd 4640	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑛⟩})
14	13	adantl 482	. . . . . . 7 ⊢ (((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) ∧ 𝑥 = 𝑛) → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑛⟩})
15		simpr 485	. . . . . . 7 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → 𝑛 ∈ 𝐵)
16		snex 5431	. . . . . . . 8 ⊢ {⟨𝑆, 𝑛⟩} ∈ V
17	16	a1i 11	. . . . . . 7 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → {⟨𝑆, 𝑛⟩} ∈ V)
18	4, 14, 15, 17	fvmptd 7005	. . . . . 6 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → (𝐹‘𝑛) = {⟨𝑆, 𝑛⟩})
19	11, 18	eqeq12d 2748	. . . . 5 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → ((𝐹‘𝑚) = (𝐹‘𝑛) ↔ {⟨𝑆, 𝑚⟩} = {⟨𝑆, 𝑛⟩}))
20	19	adantl 482	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵)) → ((𝐹‘𝑚) = (𝐹‘𝑛) ↔ {⟨𝑆, 𝑚⟩} = {⟨𝑆, 𝑛⟩}))
21		opex 5464	. . . . . 6 ⊢ ⟨𝑆, 𝑚⟩ ∈ V
22	21	sneqr 4841	. . . . 5 ⊢ ({⟨𝑆, 𝑚⟩} = {⟨𝑆, 𝑛⟩} → ⟨𝑆, 𝑚⟩ = ⟨𝑆, 𝑛⟩)
23		opthg 5477	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑚 ∈ 𝐵) → (⟨𝑆, 𝑚⟩ = ⟨𝑆, 𝑛⟩ ↔ (𝑆 = 𝑆 ∧ 𝑚 = 𝑛)))
24	23	adantrr 715	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵)) → (⟨𝑆, 𝑚⟩ = ⟨𝑆, 𝑛⟩ ↔ (𝑆 = 𝑆 ∧ 𝑚 = 𝑛)))
25		simpr 485	. . . . . 6 ⊢ ((𝑆 = 𝑆 ∧ 𝑚 = 𝑛) → 𝑚 = 𝑛)
26	24, 25	syl6bi 252	. . . . 5 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵)) → (⟨𝑆, 𝑚⟩ = ⟨𝑆, 𝑛⟩ → 𝑚 = 𝑛))
27	22, 26	syl5 34	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵)) → ({⟨𝑆, 𝑚⟩} = {⟨𝑆, 𝑛⟩} → 𝑚 = 𝑛))
28	20, 27	sylbid 239	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵)) → ((𝐹‘𝑚) = (𝐹‘𝑛) → 𝑚 = 𝑛))
29	28	ralrimivva 3200	. 2 ⊢ (𝑆 ∈ 𝑉 → ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 ((𝐹‘𝑚) = (𝐹‘𝑛) → 𝑚 = 𝑛))
30		dff13 7256	. 2 ⊢ (𝐹:𝐵–1-1→𝐴 ↔ (𝐹:𝐵⟶𝐴 ∧ ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 ((𝐹‘𝑚) = (𝐹‘𝑛) → 𝑚 = 𝑛)))
31	3, 29, 30	sylanbrc 583	1 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–1-1→𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2709  ∀wral 3061  ∃wrex 3070  Vcvv 3474  {csn 4628  ⟨cop 4634   ↦ cmpt 5231  ⟶wf 6539  –1-1→wf1 6540  ‘cfv 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fv 6551

This theorem is referenced by:  fsetsnf1o  45843