Theorem fsetsnf1 47529

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 47528	. 2 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵⟶𝐴)
4	2	a1i 11	. . . . . . 7 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → 𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩}))
5		opeq2 4808	. . . . . . . . 9 ⊢ (𝑥 = 𝑚 → ⟨𝑆, 𝑥⟩ = ⟨𝑆, 𝑚⟩)
6	5	sneqd 4570	. . . . . . . 8 ⊢ (𝑥 = 𝑚 → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑚⟩})
7	6	adantl 483	. . . . . . 7 ⊢ (((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) ∧ 𝑥 = 𝑚) → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑚⟩})
8		simpl 484	. . . . . . 7 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → 𝑚 ∈ 𝐵)
9		snex 5371	. . . . . . . 8 ⊢ {⟨𝑆, 𝑚⟩} ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → {⟨𝑆, 𝑚⟩} ∈ V)
11	4, 7, 8, 10	fvmptd 6947	. . . . . 6 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → (𝐹‘𝑚) = {⟨𝑆, 𝑚⟩})
12		opeq2 4808	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → ⟨𝑆, 𝑥⟩ = ⟨𝑆, 𝑛⟩)
13	12	sneqd 4570	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑛⟩})
14	13	adantl 483	. . . . . . 7 ⊢ (((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) ∧ 𝑥 = 𝑛) → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑛⟩})
15		simpr 486	. . . . . . 7 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → 𝑛 ∈ 𝐵)
16		snex 5371	. . . . . . . 8 ⊢ {⟨𝑆, 𝑛⟩} ∈ V
17	16	a1i 11	. . . . . . 7 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → {⟨𝑆, 𝑛⟩} ∈ V)
18	4, 14, 15, 17	fvmptd 6947	. . . . . 6 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → (𝐹‘𝑛) = {⟨𝑆, 𝑛⟩})
19	11, 18	eqeq12d 2757	. . . . 5 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → ((𝐹‘𝑚) = (𝐹‘𝑛) ↔ {⟨𝑆, 𝑚⟩} = {⟨𝑆, 𝑛⟩}))
20	19	adantl 483	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵)) → ((𝐹‘𝑚) = (𝐹‘𝑛) ↔ {⟨𝑆, 𝑚⟩} = {⟨𝑆, 𝑛⟩}))
21		opex 5406	. . . . . 6 ⊢ ⟨𝑆, 𝑚⟩ ∈ V
22	21	sneqr 4774	. . . . 5 ⊢ ({⟨𝑆, 𝑚⟩} = {⟨𝑆, 𝑛⟩} → ⟨𝑆, 𝑚⟩ = ⟨𝑆, 𝑛⟩)
23		opthg 5420	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑚 ∈ 𝐵) → (⟨𝑆, 𝑚⟩ = ⟨𝑆, 𝑛⟩ ↔ (𝑆 = 𝑆 ∧ 𝑚 = 𝑛)))
24	23	adantrr 724	. . . . . 6 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵)) → (⟨𝑆, 𝑚⟩ = ⟨𝑆, 𝑛⟩ ↔ (𝑆 = 𝑆 ∧ 𝑚 = 𝑛)))
25		simpr 486	. . . . . 6 ⊢ ((𝑆 = 𝑆 ∧ 𝑚 = 𝑛) → 𝑚 = 𝑛)
26	24, 25	biimtrdi 255	. . . . 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 242	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵)) → ((𝐹‘𝑚) = (𝐹‘𝑛) → 𝑚 = 𝑛))
29	28	ralrimivva 3184	. 2 ⊢ (𝑆 ∈ 𝑉 → ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 ((𝐹‘𝑚) = (𝐹‘𝑛) → 𝑚 = 𝑛))
30		dff13 7202	. 2 ⊢ (𝐹:𝐵–1-1→𝐴 ↔ (𝐹:𝐵⟶𝐴 ∧ ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 ((𝐹‘𝑚) = (𝐹‘𝑛) → 𝑚 = 𝑛)))
31	3, 29, 30	sylanbrc 590	1 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–1-1→𝐴)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  {cab 2719  ∀wral 3055  ∃wrex 3065  Vcvv 3433  {csn 4558  ⟨cop 4564   ↦ cmpt 5156  ⟶wf 6485  –1-1→wf1 6486  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fv 6497

This theorem is referenced by:  fsetsnf1o  47531