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Theorem fmptsng 6917
 Description: Express a singleton function in maps-to notation. Version of fmptsn 6916 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 4544 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
21bicomi 227 . . . 4 (𝑥 = 𝐴𝑥 ∈ {𝐴})
32anbi1i 626 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵))
43opabbii 5101 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵)}
5 velsn 4544 . . . . 5 (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ 𝑝 = ⟨𝐴, 𝐶⟩)
6 eqidd 2799 . . . . . . 7 ((𝐴𝑉𝐶𝑊) → 𝐴 = 𝐴)
7 eqidd 2799 . . . . . . 7 ((𝐴𝑉𝐶𝑊) → 𝐶 = 𝐶)
8 eqeq1 2802 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥 = 𝐴𝐴 = 𝐴))
98adantr 484 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐶) → (𝑥 = 𝐴𝐴 = 𝐴))
10 eqeq1 2802 . . . . . . . . . 10 (𝑦 = 𝐶 → (𝑦 = 𝐵𝐶 = 𝐵))
11 fmptsng.1 . . . . . . . . . . 11 (𝑥 = 𝐴𝐵 = 𝐶)
1211eqeq2d 2809 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐶 = 𝐵𝐶 = 𝐶))
1310, 12sylan9bbr 514 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐶) → (𝑦 = 𝐵𝐶 = 𝐶))
149, 13anbi12d 633 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐶) → ((𝑥 = 𝐴𝑦 = 𝐵) ↔ (𝐴 = 𝐴𝐶 = 𝐶)))
1514opelopabga 5389 . . . . . . 7 ((𝐴𝑉𝐶𝑊) → (⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} ↔ (𝐴 = 𝐴𝐶 = 𝐶)))
166, 7, 15mpbir2and 712 . . . . . 6 ((𝐴𝑉𝐶𝑊) → ⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)})
17 eleq1 2877 . . . . . 6 (𝑝 = ⟨𝐴, 𝐶⟩ → (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} ↔ ⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
1816, 17syl5ibrcom 250 . . . . 5 ((𝐴𝑉𝐶𝑊) → (𝑝 = ⟨𝐴, 𝐶⟩ → 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
195, 18syl5bi 245 . . . 4 ((𝐴𝑉𝐶𝑊) → (𝑝 ∈ {⟨𝐴, 𝐶⟩} → 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
20 elopab 5383 . . . . 5 (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} ↔ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴𝑦 = 𝐵)))
21 opeq12 4771 . . . . . . . . . 10 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
2221eqeq2d 2809 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑝 = ⟨𝑥, 𝑦⟩ ↔ 𝑝 = ⟨𝐴, 𝐵⟩))
2311adantr 484 . . . . . . . . . . . 12 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝐵 = 𝐶)
2423opeq2d 4776 . . . . . . . . . . 11 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐶⟩)
25 opex 5325 . . . . . . . . . . . 12 𝐴, 𝐶⟩ ∈ V
2625snid 4564 . . . . . . . . . . 11 𝐴, 𝐶⟩ ∈ {⟨𝐴, 𝐶⟩}
2724, 26eqeltrdi 2898 . . . . . . . . . 10 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐶⟩})
28 eleq1 2877 . . . . . . . . . 10 (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐶⟩}))
2927, 28syl5ibrcom 250 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑝 = ⟨𝐴, 𝐵⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
3022, 29sylbid 243 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑝 = ⟨𝑥, 𝑦⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
3130impcom 411 . . . . . . 7 ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑝 ∈ {⟨𝐴, 𝐶⟩})
3231exlimivv 1933 . . . . . 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 245 . . . 4 ((𝐴𝑉𝐶𝑊) → (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
3519, 34impbid 215 . . 3 ((𝐴𝑉𝐶𝑊) → (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
3635eqrdv 2796 . 2 ((𝐴𝑉𝐶𝑊) → {⟨𝐴, 𝐶⟩} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)})
37 df-mpt 5115 . . 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 2859 1 ((𝐴𝑉𝐶𝑊) → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {csn 4528  ⟨cop 4534  {copab 5096   ↦ cmpt 5114 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3444  df-dif 3886  df-un 3888  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-opab 5097  df-mpt 5115 This theorem is referenced by:  mdet0pr  21238  m1detdiag  21243
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