Theorem fmptsnd 6435

Description: Express a singleton function in maps-to notation. Deduction form of fmptsng 6434. (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 4193	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2	1	bicomi 214	. . . 4 ⊢ (𝑥 = 𝐴 ↔ 𝑥 ∈ {𝐴})
3	2	anbi1i 731	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵))
4	3	opabbii 4717	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵)}
5		velsn 4193	. . . . 5 ⊢ (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ 𝑝 = ⟨𝐴, 𝐶⟩)
6		eqidd 2623	. . . . . . . 8 ⊢ (𝜑 → 𝐴 = 𝐴)
7		eqidd 2623	. . . . . . . 8 ⊢ (𝜑 → 𝐶 = 𝐶)
8		sbcan 3478	. . . . . . . . . . 11 ⊢ ([𝐶 / 𝑦](𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ ([𝐶 / 𝑦]𝑥 = 𝐴 ∧ [𝐶 / 𝑦]𝑦 = 𝐵))
9		fmptsnd.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
10		sbcg 3503	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ 𝑊 → ([𝐶 / 𝑦]𝑥 = 𝐴 ↔ 𝑥 = 𝐴))
11	9, 10	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ([𝐶 / 𝑦]𝑥 = 𝐴 ↔ 𝑥 = 𝐴))
12		eqsbc3 3475	. . . . . . . . . . . . 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 747	. . . . . . . . . . 11 ⊢ (𝜑 → (([𝐶 / 𝑦]𝑥 = 𝐴 ∧ [𝐶 / 𝑦]𝑦 = 𝐵) ↔ (𝑥 = 𝐴 ∧ 𝐶 = 𝐵)))
15	8, 14	syl5bb 272	. . . . . . . . . 10 ⊢ (𝜑 → ([𝐶 / 𝑦](𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 = 𝐴 ∧ 𝐶 = 𝐵)))
16	15	sbcbidv 3490	. . . . . . . . 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 2626	. . . . . . . . . . . 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 2632	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐶 = 𝐵 ↔ 𝐶 = 𝐶))
22	19, 21	anbi12d 747	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 = 𝐴 ∧ 𝐶 = 𝐵) ↔ (𝐴 = 𝐴 ∧ 𝐶 = 𝐶)))
23	17, 22	sbcied 3472	. . . . . . . . 9 ⊢ (𝜑 → ([𝐴 / 𝑥](𝑥 = 𝐴 ∧ 𝐶 = 𝐵) ↔ (𝐴 = 𝐴 ∧ 𝐶 = 𝐶)))
24	16, 23	bitrd 268	. . . . . . . 8 ⊢ (𝜑 → ([𝐴 / 𝑥][𝐶 / 𝑦](𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝐴 = 𝐴 ∧ 𝐶 = 𝐶)))
25	6, 7, 24	mpbir2and 957	. . . . . . 7 ⊢ (𝜑 → [𝐴 / 𝑥][𝐶 / 𝑦](𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
26		opelopabsb 4985	. . . . . . 7 ⊢ (⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)} ↔ [𝐴 / 𝑥][𝐶 / 𝑦](𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
27	25, 26	sylibr 224	. . . . . 6 ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)})
28		eleq1 2689	. . . . . 6 ⊢ (𝑝 = ⟨𝐴, 𝐶⟩ → (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)} ↔ ⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)}))
29	27, 28	syl5ibrcom 237	. . . . 5 ⊢ (𝜑 → (𝑝 = ⟨𝐴, 𝐶⟩ → 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)}))
30	5, 29	syl5bi 232	. . . 4 ⊢ (𝜑 → (𝑝 ∈ {⟨𝐴, 𝐶⟩} → 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)}))
31		elopab 4983	. . . . 5 ⊢ (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)} ↔ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)))
32		opeq12 4404	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
33	32	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
34	33	eqeq2d 2632	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝑝 = ⟨𝑥, 𝑦⟩ ↔ 𝑝 = ⟨𝐴, 𝐵⟩))
35	20	adantrr 753	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝐵 = 𝐶)
36	35	opeq2d 4409	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐶⟩)
37		opex 4932	. . . . . . . . . . . . 13 ⊢ ⟨𝐴, 𝐶⟩ ∈ V
38	37	snid 4208	. . . . . . . . . . . 12 ⊢ ⟨𝐴, 𝐶⟩ ∈ {⟨𝐴, 𝐶⟩}
39	36, 38	syl6eqel 2709	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐶⟩})
40		eleq1 2689	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐶⟩}))
41	39, 40	syl5ibrcom 237	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝑝 = ⟨𝐴, 𝐵⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
42	34, 41	sylbid 230	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝑝 = ⟨𝑥, 𝑦⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
43	42	ex 450	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑝 = ⟨𝑥, 𝑦⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩})))
44	43	com23 86	. . . . . . 7 ⊢ (𝜑 → (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑝 ∈ {⟨𝐴, 𝐶⟩})))
45	44	impd 447	. . . . . 6 ⊢ (𝜑 → ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
46	45	exlimdvv 1862	. . . . 5 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
47	31, 46	syl5bi 232	. . . 4 ⊢ (𝜑 → (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)} → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
48	30, 47	impbid 202	. . 3 ⊢ (𝜑 → (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)}))
49	48	eqrdv 2620	. 2 ⊢ (𝜑 → {⟨𝐴, 𝐶⟩} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)})
50		df-mpt 4730	. . 3 ⊢ (𝑥 ∈ {𝐴} ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵)}
51	50	a1i 11	. 2 ⊢ (𝜑 → (𝑥 ∈ {𝐴} ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵)})
52	4, 49, 51	3eqtr4a 2682	1 ⊢ (𝜑 → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  [wsbc 3435  {csn 4177  ⟨cop 4183  {copab 4712   ↦ cmpt 4729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-mpt 4730

This theorem is referenced by:  fmptapd  6437  fmptpr  6438  mpt2sn  7268