Theorem fmptsng 7022

Description: Express a singleton function in maps-to notation. Version of fmptsn 7021 allowing the value 𝐵 to depend on the variable 𝑥. (Contributed by AV, 27-Feb-2019.)

Hypothesis

Ref	Expression
fmptsng.1	⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)

Assertion

Ref	Expression
fmptsng	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

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

Proof of Theorem fmptsng

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

Step	Hyp	Ref	Expression
1		velsn 4574	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2	1	bicomi 223	. . . 4 ⊢ (𝑥 = 𝐴 ↔ 𝑥 ∈ {𝐴})
3	2	anbi1i 623	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵))
4	3	opabbii 5137	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵)}
5		velsn 4574	. . . . 5 ⊢ (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ 𝑝 = ⟨𝐴, 𝐶⟩)
6		eqidd 2739	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 𝐴 = 𝐴)
7		eqidd 2739	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 𝐶 = 𝐶)
8		eqeq1 2742	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐴 ↔ 𝐴 = 𝐴))
9	8	adantr 480	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐶) → (𝑥 = 𝐴 ↔ 𝐴 = 𝐴))
10		eqeq1 2742	. . . . . . . . . 10 ⊢ (𝑦 = 𝐶 → (𝑦 = 𝐵 ↔ 𝐶 = 𝐵))
11		fmptsng.1	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
12	11	eqeq2d 2749	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝐶 = 𝐵 ↔ 𝐶 = 𝐶))
13	10, 12	sylan9bbr 510	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐶) → (𝑦 = 𝐵 ↔ 𝐶 = 𝐶))
14	9, 13	anbi12d 630	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐶) → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝐴 = 𝐴 ∧ 𝐶 = 𝐶)))
15	14	opelopabga 5439	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)} ↔ (𝐴 = 𝐴 ∧ 𝐶 = 𝐶)))
16	6, 7, 15	mpbir2and 709	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)})
17		eleq1 2826	. . . . . 6 ⊢ (𝑝 = ⟨𝐴, 𝐶⟩ → (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)} ↔ ⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)}))
18	16, 17	syl5ibrcom 246	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑝 = ⟨𝐴, 𝐶⟩ → 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)}))
19	5, 18	syl5bi 241	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑝 ∈ {⟨𝐴, 𝐶⟩} → 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)}))
20		elopab 5433	. . . . 5 ⊢ (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)} ↔ ∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)))
21		opeq12 4803	. . . . . . . . . 10 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
22	21	eqeq2d 2749	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑝 = ⟨𝑥, 𝑦⟩ ↔ 𝑝 = ⟨𝐴, 𝐵⟩))
23	11	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐵 = 𝐶)
24	23	opeq2d 4808	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐶⟩)
25		opex 5373	. . . . . . . . . . . 12 ⊢ ⟨𝐴, 𝐶⟩ ∈ V
26	25	snid 4594	. . . . . . . . . . 11 ⊢ ⟨𝐴, 𝐶⟩ ∈ {⟨𝐴, 𝐶⟩}
27	24, 26	eqeltrdi 2847	. . . . . . . . . 10 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐶⟩})
28		eleq1 2826	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐶⟩}))
29	27, 28	syl5ibrcom 246	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑝 = ⟨𝐴, 𝐵⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
30	22, 29	sylbid 239	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑝 = ⟨𝑥, 𝑦⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
31	30	impcom 407	. . . . . . 7 ⊢ ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑝 ∈ {⟨𝐴, 𝐶⟩})
32	31	exlimivv 1936	. . . . . 6 ⊢ (∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑝 ∈ {⟨𝐴, 𝐶⟩})
33	32	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (∃𝑥∃𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
34	20, 33	syl5bi 241	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)} → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
35	19, 34	impbid 211	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)}))
36	35	eqrdv 2736	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → {⟨𝐴, 𝐶⟩} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)})
37		df-mpt 5154	. . 3 ⊢ (𝑥 ∈ {𝐴} ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵)}
38	37	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑥 ∈ {𝐴} ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵)})
39	4, 36, 38	3eqtr4a 2805	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {csn 4558  ⟨cop 4564  {copab 5132   ↦ cmpt 5153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-opab 5133  df-mpt 5154

This theorem is referenced by:  mdet0pr  21649  m1detdiag  21654