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Theorem fmptsng 7188
Description: Express a singleton function in maps-to notation. Version of fmptsn 7187 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 4647 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
21bicomi 224 . . . 4 (𝑥 = 𝐴𝑥 ∈ {𝐴})
32anbi1i 624 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵))
43opabbii 5215 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵)}
5 velsn 4647 . . . . 5 (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ 𝑝 = ⟨𝐴, 𝐶⟩)
6 eqidd 2736 . . . . . . 7 ((𝐴𝑉𝐶𝑊) → 𝐴 = 𝐴)
7 eqidd 2736 . . . . . . 7 ((𝐴𝑉𝐶𝑊) → 𝐶 = 𝐶)
8 eqeq1 2739 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥 = 𝐴𝐴 = 𝐴))
98adantr 480 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐶) → (𝑥 = 𝐴𝐴 = 𝐴))
10 eqeq1 2739 . . . . . . . . . 10 (𝑦 = 𝐶 → (𝑦 = 𝐵𝐶 = 𝐵))
11 fmptsng.1 . . . . . . . . . . 11 (𝑥 = 𝐴𝐵 = 𝐶)
1211eqeq2d 2746 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐶 = 𝐵𝐶 = 𝐶))
1310, 12sylan9bbr 510 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐶) → (𝑦 = 𝐵𝐶 = 𝐶))
149, 13anbi12d 632 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐶) → ((𝑥 = 𝐴𝑦 = 𝐵) ↔ (𝐴 = 𝐴𝐶 = 𝐶)))
1514opelopabga 5543 . . . . . . 7 ((𝐴𝑉𝐶𝑊) → (⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} ↔ (𝐴 = 𝐴𝐶 = 𝐶)))
166, 7, 15mpbir2and 713 . . . . . 6 ((𝐴𝑉𝐶𝑊) → ⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)})
17 eleq1 2827 . . . . . 6 (𝑝 = ⟨𝐴, 𝐶⟩ → (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} ↔ ⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
1816, 17syl5ibrcom 247 . . . . 5 ((𝐴𝑉𝐶𝑊) → (𝑝 = ⟨𝐴, 𝐶⟩ → 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
195, 18biimtrid 242 . . . 4 ((𝐴𝑉𝐶𝑊) → (𝑝 ∈ {⟨𝐴, 𝐶⟩} → 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
20 elopab 5537 . . . . 5 (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} ↔ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴𝑦 = 𝐵)))
21 opeq12 4880 . . . . . . . . . 10 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
2221eqeq2d 2746 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑝 = ⟨𝑥, 𝑦⟩ ↔ 𝑝 = ⟨𝐴, 𝐵⟩))
2311adantr 480 . . . . . . . . . . . 12 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝐵 = 𝐶)
2423opeq2d 4885 . . . . . . . . . . 11 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐶⟩)
25 opex 5475 . . . . . . . . . . . 12 𝐴, 𝐶⟩ ∈ V
2625snid 4667 . . . . . . . . . . 11 𝐴, 𝐶⟩ ∈ {⟨𝐴, 𝐶⟩}
2724, 26eqeltrdi 2847 . . . . . . . . . 10 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐶⟩})
28 eleq1 2827 . . . . . . . . . 10 (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐶⟩}))
2927, 28syl5ibrcom 247 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑝 = ⟨𝐴, 𝐵⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
3022, 29sylbid 240 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑝 = ⟨𝑥, 𝑦⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
3130impcom 407 . . . . . . 7 ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑝 ∈ {⟨𝐴, 𝐶⟩})
3231exlimivv 1930 . . . . . 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, 33biimtrid 242 . . . 4 ((𝐴𝑉𝐶𝑊) → (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
3519, 34impbid 212 . . 3 ((𝐴𝑉𝐶𝑊) → (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
3635eqrdv 2733 . 2 ((𝐴𝑉𝐶𝑊) → {⟨𝐴, 𝐶⟩} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)})
37 df-mpt 5232 . . 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 2801 1 ((𝐴𝑉𝐶𝑊) → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  {csn 4631  cop 4637  {copab 5210  cmpt 5231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-mpt 5232
This theorem is referenced by:  mdet0pr  22614  m1detdiag  22619
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