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Theorem fmptsng 6475
Description: Express a singleton function in maps-to notation. Version of fmptsn 6474 allowing the value 𝐵 to depend on the variable 𝑥. (Contributed by AV, 27-Feb-2019.)
Hypothesis
Ref Expression
fmptsng.1 (𝑥 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
fmptsng ((𝐴𝑉𝐶𝑊) → {⟨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.
StepHypRef Expression
1 velsn 4226 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
21bicomi 214 . . . 4 (𝑥 = 𝐴𝑥 ∈ {𝐴})
32anbi1i 731 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵))
43opabbii 4750 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵)}
5 velsn 4226 . . . . 5 (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ 𝑝 = ⟨𝐴, 𝐶⟩)
6 eqidd 2652 . . . . . . 7 ((𝐴𝑉𝐶𝑊) → 𝐴 = 𝐴)
7 eqidd 2652 . . . . . . 7 ((𝐴𝑉𝐶𝑊) → 𝐶 = 𝐶)
8 eqeq1 2655 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥 = 𝐴𝐴 = 𝐴))
98adantr 480 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐶) → (𝑥 = 𝐴𝐴 = 𝐴))
10 eqeq1 2655 . . . . . . . . . 10 (𝑦 = 𝐶 → (𝑦 = 𝐵𝐶 = 𝐵))
11 fmptsng.1 . . . . . . . . . . 11 (𝑥 = 𝐴𝐵 = 𝐶)
1211eqeq2d 2661 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐶 = 𝐵𝐶 = 𝐶))
1310, 12sylan9bbr 737 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐶) → (𝑦 = 𝐵𝐶 = 𝐶))
149, 13anbi12d 747 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐶) → ((𝑥 = 𝐴𝑦 = 𝐵) ↔ (𝐴 = 𝐴𝐶 = 𝐶)))
1514opelopabga 5017 . . . . . . 7 ((𝐴𝑉𝐶𝑊) → (⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} ↔ (𝐴 = 𝐴𝐶 = 𝐶)))
166, 7, 15mpbir2and 977 . . . . . 6 ((𝐴𝑉𝐶𝑊) → ⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)})
17 eleq1 2718 . . . . . 6 (𝑝 = ⟨𝐴, 𝐶⟩ → (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} ↔ ⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
1816, 17syl5ibrcom 237 . . . . 5 ((𝐴𝑉𝐶𝑊) → (𝑝 = ⟨𝐴, 𝐶⟩ → 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
195, 18syl5bi 232 . . . 4 ((𝐴𝑉𝐶𝑊) → (𝑝 ∈ {⟨𝐴, 𝐶⟩} → 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
20 elopab 5012 . . . . 5 (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} ↔ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴𝑦 = 𝐵)))
21 opeq12 4435 . . . . . . . . . 10 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
2221eqeq2d 2661 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑝 = ⟨𝑥, 𝑦⟩ ↔ 𝑝 = ⟨𝐴, 𝐵⟩))
2311adantr 480 . . . . . . . . . . . 12 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝐵 = 𝐶)
2423opeq2d 4440 . . . . . . . . . . 11 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐶⟩)
25 opex 4962 . . . . . . . . . . . 12 𝐴, 𝐶⟩ ∈ V
2625snid 4241 . . . . . . . . . . 11 𝐴, 𝐶⟩ ∈ {⟨𝐴, 𝐶⟩}
2724, 26syl6eqel 2738 . . . . . . . . . 10 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐶⟩})
28 eleq1 2718 . . . . . . . . . 10 (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐶⟩}))
2927, 28syl5ibrcom 237 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑝 = ⟨𝐴, 𝐵⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
3022, 29sylbid 230 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑝 = ⟨𝑥, 𝑦⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
3130impcom 445 . . . . . . 7 ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑝 ∈ {⟨𝐴, 𝐶⟩})
3231exlimivv 1900 . . . . . 6 (∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑝 ∈ {⟨𝐴, 𝐶⟩})
3332a1i 11 . . . . 5 ((𝐴𝑉𝐶𝑊) → (∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
3420, 33syl5bi 232 . . . 4 ((𝐴𝑉𝐶𝑊) → (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
3519, 34impbid 202 . . 3 ((𝐴𝑉𝐶𝑊) → (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
3635eqrdv 2649 . 2 ((𝐴𝑉𝐶𝑊) → {⟨𝐴, 𝐶⟩} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)})
37 df-mpt 4763 . . 3 (𝑥 ∈ {𝐴} ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵)}
3837a1i 11 . 2 ((𝐴𝑉𝐶𝑊) → (𝑥 ∈ {𝐴} ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵)})
394, 36, 383eqtr4a 2711 1 ((𝐴𝑉𝐶𝑊) → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  {csn 4210  cop 4216  {copab 4745  cmpt 4762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-mpt 4763
This theorem is referenced by:  mdet0pr  20446  m1detdiag  20451
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