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

Theorem fmptsnd 6420
Description: Express a singleton function in maps-to notation. Deduction form of fmptsng 6419. (Contributed by AV, 4-Aug-2019.)
Hypotheses
Ref Expression
fmptsnd.1 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
fmptsnd.2 (𝜑𝐴𝑉)
fmptsnd.3 (𝜑𝐶𝑊)
Assertion
Ref Expression
fmptsnd (𝜑 → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem fmptsnd
Dummy variables 𝑝 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 velsn 4184 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
21bicomi 214 . . . 4 (𝑥 = 𝐴𝑥 ∈ {𝐴})
32anbi1i 730 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵))
43opabbii 4708 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵)}
5 velsn 4184 . . . . 5 (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ 𝑝 = ⟨𝐴, 𝐶⟩)
6 eqidd 2621 . . . . . . . 8 (𝜑𝐴 = 𝐴)
7 eqidd 2621 . . . . . . . 8 (𝜑𝐶 = 𝐶)
8 sbcan 3472 . . . . . . . . . . 11 ([𝐶 / 𝑦](𝑥 = 𝐴𝑦 = 𝐵) ↔ ([𝐶 / 𝑦]𝑥 = 𝐴[𝐶 / 𝑦]𝑦 = 𝐵))
9 fmptsnd.3 . . . . . . . . . . . . 13 (𝜑𝐶𝑊)
10 sbcg 3497 . . . . . . . . . . . . 13 (𝐶𝑊 → ([𝐶 / 𝑦]𝑥 = 𝐴𝑥 = 𝐴))
119, 10syl 17 . . . . . . . . . . . 12 (𝜑 → ([𝐶 / 𝑦]𝑥 = 𝐴𝑥 = 𝐴))
12 eqsbc3 3469 . . . . . . . . . . . . 13 (𝐶𝑊 → ([𝐶 / 𝑦]𝑦 = 𝐵𝐶 = 𝐵))
139, 12syl 17 . . . . . . . . . . . 12 (𝜑 → ([𝐶 / 𝑦]𝑦 = 𝐵𝐶 = 𝐵))
1411, 13anbi12d 746 . . . . . . . . . . 11 (𝜑 → (([𝐶 / 𝑦]𝑥 = 𝐴[𝐶 / 𝑦]𝑦 = 𝐵) ↔ (𝑥 = 𝐴𝐶 = 𝐵)))
158, 14syl5bb 272 . . . . . . . . . 10 (𝜑 → ([𝐶 / 𝑦](𝑥 = 𝐴𝑦 = 𝐵) ↔ (𝑥 = 𝐴𝐶 = 𝐵)))
1615sbcbidv 3484 . . . . . . . . 9 (𝜑 → ([𝐴 / 𝑥][𝐶 / 𝑦](𝑥 = 𝐴𝑦 = 𝐵) ↔ [𝐴 / 𝑥](𝑥 = 𝐴𝐶 = 𝐵)))
17 fmptsnd.2 . . . . . . . . . 10 (𝜑𝐴𝑉)
18 eqeq1 2624 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (𝑥 = 𝐴𝐴 = 𝐴))
1918adantl 482 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐴) → (𝑥 = 𝐴𝐴 = 𝐴))
20 fmptsnd.1 . . . . . . . . . . . 12 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
2120eqeq2d 2630 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐴) → (𝐶 = 𝐵𝐶 = 𝐶))
2219, 21anbi12d 746 . . . . . . . . . 10 ((𝜑𝑥 = 𝐴) → ((𝑥 = 𝐴𝐶 = 𝐵) ↔ (𝐴 = 𝐴𝐶 = 𝐶)))
2317, 22sbcied 3466 . . . . . . . . 9 (𝜑 → ([𝐴 / 𝑥](𝑥 = 𝐴𝐶 = 𝐵) ↔ (𝐴 = 𝐴𝐶 = 𝐶)))
2416, 23bitrd 268 . . . . . . . 8 (𝜑 → ([𝐴 / 𝑥][𝐶 / 𝑦](𝑥 = 𝐴𝑦 = 𝐵) ↔ (𝐴 = 𝐴𝐶 = 𝐶)))
256, 7, 24mpbir2and 956 . . . . . . 7 (𝜑[𝐴 / 𝑥][𝐶 / 𝑦](𝑥 = 𝐴𝑦 = 𝐵))
26 opelopabsb 4975 . . . . . . 7 (⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} ↔ [𝐴 / 𝑥][𝐶 / 𝑦](𝑥 = 𝐴𝑦 = 𝐵))
2725, 26sylibr 224 . . . . . 6 (𝜑 → ⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)})
28 eleq1 2687 . . . . . 6 (𝑝 = ⟨𝐴, 𝐶⟩ → (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} ↔ ⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
2927, 28syl5ibrcom 237 . . . . 5 (𝜑 → (𝑝 = ⟨𝐴, 𝐶⟩ → 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
305, 29syl5bi 232 . . . 4 (𝜑 → (𝑝 ∈ {⟨𝐴, 𝐶⟩} → 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
31 elopab 4973 . . . . 5 (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} ↔ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴𝑦 = 𝐵)))
32 opeq12 4395 . . . . . . . . . . . 12 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
3332adantl 482 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
3433eqeq2d 2630 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝑝 = ⟨𝑥, 𝑦⟩ ↔ 𝑝 = ⟨𝐴, 𝐵⟩))
3520adantrr 752 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝐵 = 𝐶)
3635opeq2d 4400 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐶⟩)
37 opex 4923 . . . . . . . . . . . . 13 𝐴, 𝐶⟩ ∈ V
3837snid 4199 . . . . . . . . . . . 12 𝐴, 𝐶⟩ ∈ {⟨𝐴, 𝐶⟩}
3936, 38syl6eqel 2707 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐶⟩})
40 eleq1 2687 . . . . . . . . . . 11 (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐶⟩}))
4139, 40syl5ibrcom 237 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝑝 = ⟨𝐴, 𝐵⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
4234, 41sylbid 230 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝑝 = ⟨𝑥, 𝑦⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
4342ex 450 . . . . . . . 8 (𝜑 → ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑝 = ⟨𝑥, 𝑦⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩})))
4443com23 86 . . . . . . 7 (𝜑 → (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑝 ∈ {⟨𝐴, 𝐶⟩})))
4544impd 447 . . . . . 6 (𝜑 → ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
4645exlimdvv 1860 . . . . 5 (𝜑 → (∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
4731, 46syl5bi 232 . . . 4 (𝜑 → (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
4830, 47impbid 202 . . 3 (𝜑 → (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
4948eqrdv 2618 . 2 (𝜑 → {⟨𝐴, 𝐶⟩} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)})
50 df-mpt 4721 . . 3 (𝑥 ∈ {𝐴} ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵)}
5150a1i 11 . 2 (𝜑 → (𝑥 ∈ {𝐴} ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵)})
524, 49, 513eqtr4a 2680 1 (𝜑 → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wex 1702  wcel 1988  [wsbc 3429  {csn 4168  cop 4174  {copab 4703  cmpt 4720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-opab 4704  df-mpt 4721
This theorem is referenced by:  fmptapd  6422  fmptpr  6423  mpt2sn  7253
  Copyright terms: Public domain W3C validator