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Theorem fmptsnd 6918
 Description: Express a singleton function in maps-to notation. Deduction form of fmptsng 6917. (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 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 . . . . . . . 8 (𝜑𝐴 = 𝐴)
7 eqidd 2799 . . . . . . . 8 (𝜑𝐶 = 𝐶)
8 sbcan 3770 . . . . . . . . . . 11 ([𝐶 / 𝑦](𝑥 = 𝐴𝑦 = 𝐵) ↔ ([𝐶 / 𝑦]𝑥 = 𝐴[𝐶 / 𝑦]𝑦 = 𝐵))
9 fmptsnd.3 . . . . . . . . . . . . 13 (𝜑𝐶𝑊)
10 sbcg 3795 . . . . . . . . . . . . 13 (𝐶𝑊 → ([𝐶 / 𝑦]𝑥 = 𝐴𝑥 = 𝐴))
119, 10syl 17 . . . . . . . . . . . 12 (𝜑 → ([𝐶 / 𝑦]𝑥 = 𝐴𝑥 = 𝐴))
12 eqsbc3 3767 . . . . . . . . . . . . 13 (𝐶𝑊 → ([𝐶 / 𝑦]𝑦 = 𝐵𝐶 = 𝐵))
139, 12syl 17 . . . . . . . . . . . 12 (𝜑 → ([𝐶 / 𝑦]𝑦 = 𝐵𝐶 = 𝐵))
1411, 13anbi12d 633 . . . . . . . . . . 11 (𝜑 → (([𝐶 / 𝑦]𝑥 = 𝐴[𝐶 / 𝑦]𝑦 = 𝐵) ↔ (𝑥 = 𝐴𝐶 = 𝐵)))
158, 14syl5bb 286 . . . . . . . . . 10 (𝜑 → ([𝐶 / 𝑦](𝑥 = 𝐴𝑦 = 𝐵) ↔ (𝑥 = 𝐴𝐶 = 𝐵)))
1615sbcbidv 3776 . . . . . . . . 9 (𝜑 → ([𝐴 / 𝑥][𝐶 / 𝑦](𝑥 = 𝐴𝑦 = 𝐵) ↔ [𝐴 / 𝑥](𝑥 = 𝐴𝐶 = 𝐵)))
17 fmptsnd.2 . . . . . . . . . 10 (𝜑𝐴𝑉)
18 eqeq1 2802 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (𝑥 = 𝐴𝐴 = 𝐴))
1918adantl 485 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐴) → (𝑥 = 𝐴𝐴 = 𝐴))
20 fmptsnd.1 . . . . . . . . . . . 12 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
2120eqeq2d 2809 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐴) → (𝐶 = 𝐵𝐶 = 𝐶))
2219, 21anbi12d 633 . . . . . . . . . 10 ((𝜑𝑥 = 𝐴) → ((𝑥 = 𝐴𝐶 = 𝐵) ↔ (𝐴 = 𝐴𝐶 = 𝐶)))
2317, 22sbcied 3764 . . . . . . . . 9 (𝜑 → ([𝐴 / 𝑥](𝑥 = 𝐴𝐶 = 𝐵) ↔ (𝐴 = 𝐴𝐶 = 𝐶)))
2416, 23bitrd 282 . . . . . . . 8 (𝜑 → ([𝐴 / 𝑥][𝐶 / 𝑦](𝑥 = 𝐴𝑦 = 𝐵) ↔ (𝐴 = 𝐴𝐶 = 𝐶)))
256, 7, 24mpbir2and 712 . . . . . . 7 (𝜑[𝐴 / 𝑥][𝐶 / 𝑦](𝑥 = 𝐴𝑦 = 𝐵))
26 opelopabsb 5386 . . . . . . 7 (⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} ↔ [𝐴 / 𝑥][𝐶 / 𝑦](𝑥 = 𝐴𝑦 = 𝐵))
2725, 26sylibr 237 . . . . . 6 (𝜑 → ⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)})
28 eleq1 2877 . . . . . 6 (𝑝 = ⟨𝐴, 𝐶⟩ → (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} ↔ ⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
2927, 28syl5ibrcom 250 . . . . 5 (𝜑 → (𝑝 = ⟨𝐴, 𝐶⟩ → 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
305, 29syl5bi 245 . . . 4 (𝜑 → (𝑝 ∈ {⟨𝐴, 𝐶⟩} → 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
31 elopab 5383 . . . . 5 (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} ↔ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴𝑦 = 𝐵)))
32 opeq12 4771 . . . . . . . . . . 11 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
3332adantl 485 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
3433eqeq2d 2809 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝑝 = ⟨𝑥, 𝑦⟩ ↔ 𝑝 = ⟨𝐴, 𝐵⟩))
3520adantrr 716 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝐵 = 𝐶)
3635opeq2d 4776 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐶⟩)
37 opex 5325 . . . . . . . . . . . 12 𝐴, 𝐶⟩ ∈ V
3837snid 4564 . . . . . . . . . . 11 𝐴, 𝐶⟩ ∈ {⟨𝐴, 𝐶⟩}
3936, 38eqeltrdi 2898 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐶⟩})
40 eleq1 2877 . . . . . . . . . 10 (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐶⟩}))
4139, 40syl5ibrcom 250 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝑝 = ⟨𝐴, 𝐵⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
4234, 41sylbid 243 . . . . . . . 8 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝑝 = ⟨𝑥, 𝑦⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
4342ex 416 . . . . . . 7 (𝜑 → ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑝 = ⟨𝑥, 𝑦⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩})))
4443impcomd 415 . . . . . 6 (𝜑 → ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
4544exlimdvv 1935 . . . . 5 (𝜑 → (∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
4631, 45syl5bi 245 . . . 4 (𝜑 → (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
4730, 46impbid 215 . . 3 (𝜑 → (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
4847eqrdv 2796 . 2 (𝜑 → {⟨𝐴, 𝐶⟩} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)})
49 df-mpt 5115 . . 3 (𝑥 ∈ {𝐴} ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵)}
5049a1i 11 . 2 (𝜑 → (𝑥 ∈ {𝐴} ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵)})
514, 48, 503eqtr4a 2859 1 (𝜑 → {⟨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  [wsbc 3722  {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-ne 2988  df-ral 3111  df-rex 3112  df-v 3444  df-sbc 3723  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:  fmptapd  6920  fmptpr  6921  mposn  7794
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