Theorem fmptsnd 7041

Description: Express a singleton function in maps-to notation. Deduction form of fmptsng 7040. (Contributed by AV, 4-Aug-2019.)

Hypotheses

Ref	Expression
fmptsnd.1	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)
fmptsnd.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fmptsnd.3	⊢ (𝜑 → 𝐶 ∈ 𝑊)

Assertion

Ref	Expression
fmptsnd	⊢ (𝜑 → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝜑,𝑥

Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem fmptsnd

Dummy variables 𝑝 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		velsn 4577	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2	1	bicomi 223	. . . 4 ⊢ (𝑥 = 𝐴 ↔ 𝑥 ∈ {𝐴})
3	2	anbi1i 624	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵))
4	3	opabbii 5141	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵)}
5		velsn 4577	. . . . 5 ⊢ (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ 𝑝 = ⟨𝐴, 𝐶⟩)
6		eqidd 2739	. . . . . . . 8 ⊢ (𝜑 → 𝐴 = 𝐴)
7		eqidd 2739	. . . . . . . 8 ⊢ (𝜑 → 𝐶 = 𝐶)
8		sbcan 3768	. . . . . . . . . . 11 ⊢ ([𝐶 / 𝑦](𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ([𝐶 / 𝑦]𝑥 = 𝐴 ∧ [𝐶 / 𝑦]𝑦 = 𝐵))
9		fmptsnd.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
10		sbcg 3795	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ 𝑊 → ([𝐶 / 𝑦]𝑥 = 𝐴 ↔ 𝑥 = 𝐴))
11	9, 10	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ([𝐶 / 𝑦]𝑥 = 𝐴 ↔ 𝑥 = 𝐴))
12		eqsbc1 3765	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ 𝑊 → ([𝐶 / 𝑦]𝑦 = 𝐵 ↔ 𝐶 = 𝐵))
13	9, 12	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ([𝐶 / 𝑦]𝑦 = 𝐵 ↔ 𝐶 = 𝐵))
14	11, 13	anbi12d 631	. . . . . . . . . . 11 ⊢ (𝜑 → (([𝐶 / 𝑦]𝑥 = 𝐴 ∧ [𝐶 / 𝑦]𝑦 = 𝐵) ↔ (𝑥 = 𝐴 ∧ 𝐶 = 𝐵)))
15	8, 14	bitrid 282	. . . . . . . . . 10 ⊢ (𝜑 → ([𝐶 / 𝑦](𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 = 𝐴 ∧ 𝐶 = 𝐵)))
16	15	sbcbidv 3775	. . . . . . . . 9 ⊢ (𝜑 → ([𝐴 / 𝑥][𝐶 / 𝑦](𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ [𝐴 / 𝑥](𝑥 = 𝐴 ∧ 𝐶 = 𝐵)))
17		fmptsnd.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
18		eqeq1 2742	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐴 ↔ 𝐴 = 𝐴))
19	18	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑥 = 𝐴 ↔ 𝐴 = 𝐴))
20		fmptsnd.1	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)
21	20	eqeq2d 2749	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐶 = 𝐵 ↔ 𝐶 = 𝐶))
22	19, 21	anbi12d 631	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 = 𝐴 ∧ 𝐶 = 𝐵) ↔ (𝐴 = 𝐴 ∧ 𝐶 = 𝐶)))
23	17, 22	sbcied 3761	. . . . . . . . 9 ⊢ (𝜑 → ([𝐴 / 𝑥](𝑥 = 𝐴 ∧ 𝐶 = 𝐵) ↔ (𝐴 = 𝐴 ∧ 𝐶 = 𝐶)))
24	16, 23	bitrd 278	. . . . . . . 8 ⊢ (𝜑 → ([𝐴 / 𝑥][𝐶 / 𝑦](𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝐴 = 𝐴 ∧ 𝐶 = 𝐶)))
25	6, 7, 24	mpbir2and 710	. . . . . . 7 ⊢ (𝜑 → [𝐴 / 𝑥][𝐶 / 𝑦](𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
26		opelopabsb 5443	. . . . . . 7 ⊢ (⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)} ↔ [𝐴 / 𝑥][𝐶 / 𝑦](𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
27	25, 26	sylibr 233	. . . . . 6 ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)})
28		eleq1 2826	. . . . . 6 ⊢ (𝑝 = ⟨𝐴, 𝐶⟩ → (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)} ↔ ⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)}))
29	27, 28	syl5ibrcom 246	. . . . 5 ⊢ (𝜑 → (𝑝 = ⟨𝐴, 𝐶⟩ → 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)}))
30	5, 29	syl5bi 241	. . . 4 ⊢ (𝜑 → (𝑝 ∈ {⟨𝐴, 𝐶⟩} → 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)}))
31		elopab 5440	. . . . 5 ⊢ (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)} ↔ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)))
32		opeq12 4806	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
33	32	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
34	33	eqeq2d 2749	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝑝 = ⟨𝑥, 𝑦⟩ ↔ 𝑝 = ⟨𝐴, 𝐵⟩))
35	20	adantrr 714	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝐵 = 𝐶)
36	35	opeq2d 4811	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐶⟩)
37		opex 5379	. . . . . . . . . . . 12 ⊢ ⟨𝐴, 𝐶⟩ ∈ V
38	37	snid 4597	. . . . . . . . . . 11 ⊢ ⟨𝐴, 𝐶⟩ ∈ {⟨𝐴, 𝐶⟩}
39	36, 38	eqeltrdi 2847	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐶⟩})
40		eleq1 2826	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐶⟩}))
41	39, 40	syl5ibrcom 246	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝑝 = ⟨𝐴, 𝐵⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
42	34, 41	sylbid 239	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝑝 = ⟨𝑥, 𝑦⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
43	42	ex 413	. . . . . . 7 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑝 = ⟨𝑥, 𝑦⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩})))
44	43	impcomd 412	. . . . . 6 ⊢ (𝜑 → ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
45	44	exlimdvv 1937	. . . . 5 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
46	31, 45	syl5bi 241	. . . 4 ⊢ (𝜑 → (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)} → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
47	30, 46	impbid 211	. . 3 ⊢ (𝜑 → (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)}))
48	47	eqrdv 2736	. 2 ⊢ (𝜑 → {⟨𝐴, 𝐶⟩} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)})
49		df-mpt 5158	. . 3 ⊢ (𝑥 ∈ {𝐴} ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵)}
50	49	a1i 11	. 2 ⊢ (𝜑 → (𝑥 ∈ {𝐴} ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵)})
51	4, 48, 50	3eqtr4a 2804	1 ⊢ (𝜑 → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  [wsbc 3716  {csn 4561  ⟨cop 4567  {copab 5136   ↦ cmpt 5157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5137  df-mpt 5158

This theorem is referenced by:  fmptapd  7043  fmptpr  7044  mposn  7943