MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fmptsng Structured version   Visualization version   GIF version

Theorem fmptsng 6586
Description: Express a singleton function in maps-to notation. Version of fmptsn 6585 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 4325 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
21bicomi 214 . . . 4 (𝑥 = 𝐴𝑥 ∈ {𝐴})
32anbi1i 733 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵))
43opabbii 4857 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵)}
5 velsn 4325 . . . . 5 (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ 𝑝 = ⟨𝐴, 𝐶⟩)
6 eqidd 2749 . . . . . . 7 ((𝐴𝑉𝐶𝑊) → 𝐴 = 𝐴)
7 eqidd 2749 . . . . . . 7 ((𝐴𝑉𝐶𝑊) → 𝐶 = 𝐶)
8 eqeq1 2752 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥 = 𝐴𝐴 = 𝐴))
98adantr 472 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐶) → (𝑥 = 𝐴𝐴 = 𝐴))
10 eqeq1 2752 . . . . . . . . . 10 (𝑦 = 𝐶 → (𝑦 = 𝐵𝐶 = 𝐵))
11 fmptsng.1 . . . . . . . . . . 11 (𝑥 = 𝐴𝐵 = 𝐶)
1211eqeq2d 2758 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐶 = 𝐵𝐶 = 𝐶))
1310, 12sylan9bbr 739 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐶) → (𝑦 = 𝐵𝐶 = 𝐶))
149, 13anbi12d 749 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐶) → ((𝑥 = 𝐴𝑦 = 𝐵) ↔ (𝐴 = 𝐴𝐶 = 𝐶)))
1514opelopabga 5126 . . . . . . 7 ((𝐴𝑉𝐶𝑊) → (⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} ↔ (𝐴 = 𝐴𝐶 = 𝐶)))
166, 7, 15mpbir2and 995 . . . . . 6 ((𝐴𝑉𝐶𝑊) → ⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)})
17 eleq1 2815 . . . . . 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 5121 . . . . 5 (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} ↔ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴𝑦 = 𝐵)))
21 opeq12 4543 . . . . . . . . . 10 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
2221eqeq2d 2758 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑝 = ⟨𝑥, 𝑦⟩ ↔ 𝑝 = ⟨𝐴, 𝐵⟩))
2311adantr 472 . . . . . . . . . . . 12 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝐵 = 𝐶)
2423opeq2d 4548 . . . . . . . . . . 11 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐶⟩)
25 opex 5069 . . . . . . . . . . . 12 𝐴, 𝐶⟩ ∈ V
2625snid 4341 . . . . . . . . . . 11 𝐴, 𝐶⟩ ∈ {⟨𝐴, 𝐶⟩}
2724, 26syl6eqel 2835 . . . . . . . . . 10 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐶⟩})
28 eleq1 2815 . . . . . . . . . 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 1997 . . . . . 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 2746 . 2 ((𝐴𝑉𝐶𝑊) → {⟨𝐴, 𝐶⟩} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)})
37 df-mpt 4870 . . 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 2808 1 ((𝐴𝑉𝐶𝑊) → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1620  wex 1841  wcel 2127  {csn 4309  cop 4315  {copab 4852  cmpt 4869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-rab 3047  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-opab 4853  df-mpt 4870
This theorem is referenced by:  mdet0pr  20571  m1detdiag  20576
  Copyright terms: Public domain W3C validator