Theorem fsetsnf1 47941

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 47940	. 2 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵⟶𝐴)
4	2	a1i 11	. . . . . . 7 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → 𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩}))
5		opeq2 4834	. . . . . . . . 9 ⊢ (𝑥 = 𝑚 → ⟨𝑆, 𝑥⟩ = ⟨𝑆, 𝑚⟩)
6	5	sneqd 4596	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑚⟩})
7	6	adantl 487	. . . . . . 7 ⊢ (((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) ∧ 𝑥 = 𝑚) → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑚⟩})
8		simpl 488	. . . . . . 7 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → 𝑚 ∈ 𝐵)
9		snex 5404	. . . . . . . 8 ⊢ {⟨𝑆, 𝑚⟩} ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → {⟨𝑆, 𝑚⟩} ∈ V)
11	4, 7, 8, 10	fvmptd 6995	. . . . . 6 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → (𝐹‘𝑚) = {⟨𝑆, 𝑚⟩})
12		opeq2 4834	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → ⟨𝑆, 𝑥⟩ = ⟨𝑆, 𝑛⟩)
13	12	sneqd 4596	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑛⟩})
14	13	adantl 487	. . . . . . 7 ⊢ (((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) ∧ 𝑥 = 𝑛) → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑛⟩})
15		simpr 490	. . . . . . 7 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → 𝑛 ∈ 𝐵)
16		snex 5404	. . . . . . . 8 ⊢ {⟨𝑆, 𝑛⟩} ∈ V
17	16	a1i 11	. . . . . . 7 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → {⟨𝑆, 𝑛⟩} ∈ V)
18	4, 14, 15, 17	fvmptd 6995	. . . . . 6 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → (𝐹‘𝑛) = {⟨𝑆, 𝑛⟩})
19	11, 18	eqeq12d 2776	. . . . 5 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → ((𝐹‘𝑚) = (𝐹‘𝑛) ↔ {⟨𝑆, 𝑚⟩} = {⟨𝑆, 𝑛⟩}))
20	19	adantl 487	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵)) → ((𝐹‘𝑚) = (𝐹‘𝑛) ↔ {⟨𝑆, 𝑚⟩} = {⟨𝑆, 𝑛⟩}))
21		opex 5439	. . . . . 6 ⊢ ⟨𝑆, 𝑚⟩ ∈ V
22	21	sneqr 4800	. . . . 5 ⊢ ({⟨𝑆, 𝑚⟩} = {⟨𝑆, 𝑛⟩} → ⟨𝑆, 𝑚⟩ = ⟨𝑆, 𝑛⟩)
23		opthg 5453	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑚 ∈ 𝐵) → (⟨𝑆, 𝑚⟩ = ⟨𝑆, 𝑛⟩ ↔ (𝑆 = 𝑆 ∧ 𝑚 = 𝑛)))
24	23	adantrr 730	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵)) → (⟨𝑆, 𝑚⟩ = ⟨𝑆, 𝑛⟩ ↔ (𝑆 = 𝑆 ∧ 𝑚 = 𝑛)))
25		simpr 490	. . . . . 6 ⊢ ((𝑆 = 𝑆 ∧ 𝑚 = 𝑛) → 𝑚 = 𝑛)
26	24, 25	biimtrdi 256	. . . . 5 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵)) → (⟨𝑆, 𝑚⟩ = ⟨𝑆, 𝑛⟩ → 𝑚 = 𝑛))
27	22, 26	syl5 35	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵)) → ({⟨𝑆, 𝑚⟩} = {⟨𝑆, 𝑛⟩} → 𝑚 = 𝑛))
28	20, 27	sylbid 243	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵)) → ((𝐹‘𝑚) = (𝐹‘𝑛) → 𝑚 = 𝑛))
29	28	ralrimivva 3205	. 2 ⊢ (𝑆 ∈ 𝑉 → ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 ((𝐹‘𝑚) = (𝐹‘𝑛) → 𝑚 = 𝑛))
30		dff13 7252	. 2 ⊢ (𝐹:𝐵–1-1→𝐴 ↔ (𝐹:𝐵⟶𝐴 ∧ ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 ((𝐹‘𝑚) = (𝐹‘𝑛) → 𝑚 = 𝑛)))
31	3, 29, 30	sylanbrc 595	1 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–1-1→𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃wrex 3086  Vcvv 3450  {csn 4584  ⟨cop 4590   ↦ cmpt 5186  ⟶wf 6529  –1-1→wf1 6530  ‘cfv 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fv 6541

This theorem is used by:  fsetsnf1o  47943