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Theorem fmptsnd 6794
Description: Express a singleton function in maps-to notation. Deduction form of fmptsng 6793. (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 4488 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
21bicomi 225 . . . 4 (𝑥 = 𝐴𝑥 ∈ {𝐴})
32anbi1i 623 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵))
43opabbii 5029 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝐴} ∧ 𝑦 = 𝐵)}
5 velsn 4488 . . . . 5 (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ 𝑝 = ⟨𝐴, 𝐶⟩)
6 eqidd 2796 . . . . . . . 8 (𝜑𝐴 = 𝐴)
7 eqidd 2796 . . . . . . . 8 (𝜑𝐶 = 𝐶)
8 sbcan 3750 . . . . . . . . . . 11 ([𝐶 / 𝑦](𝑥 = 𝐴𝑦 = 𝐵) ↔ ([𝐶 / 𝑦]𝑥 = 𝐴[𝐶 / 𝑦]𝑦 = 𝐵))
9 fmptsnd.3 . . . . . . . . . . . . 13 (𝜑𝐶𝑊)
10 sbcg 3775 . . . . . . . . . . . . 13 (𝐶𝑊 → ([𝐶 / 𝑦]𝑥 = 𝐴𝑥 = 𝐴))
119, 10syl 17 . . . . . . . . . . . 12 (𝜑 → ([𝐶 / 𝑦]𝑥 = 𝐴𝑥 = 𝐴))
12 eqsbc3 3746 . . . . . . . . . . . . 13 (𝐶𝑊 → ([𝐶 / 𝑦]𝑦 = 𝐵𝐶 = 𝐵))
139, 12syl 17 . . . . . . . . . . . 12 (𝜑 → ([𝐶 / 𝑦]𝑦 = 𝐵𝐶 = 𝐵))
1411, 13anbi12d 630 . . . . . . . . . . 11 (𝜑 → (([𝐶 / 𝑦]𝑥 = 𝐴[𝐶 / 𝑦]𝑦 = 𝐵) ↔ (𝑥 = 𝐴𝐶 = 𝐵)))
158, 14syl5bb 284 . . . . . . . . . 10 (𝜑 → ([𝐶 / 𝑦](𝑥 = 𝐴𝑦 = 𝐵) ↔ (𝑥 = 𝐴𝐶 = 𝐵)))
1615sbcbidv 3756 . . . . . . . . 9 (𝜑 → ([𝐴 / 𝑥][𝐶 / 𝑦](𝑥 = 𝐴𝑦 = 𝐵) ↔ [𝐴 / 𝑥](𝑥 = 𝐴𝐶 = 𝐵)))
17 fmptsnd.2 . . . . . . . . . 10 (𝜑𝐴𝑉)
18 eqeq1 2799 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (𝑥 = 𝐴𝐴 = 𝐴))
1918adantl 482 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐴) → (𝑥 = 𝐴𝐴 = 𝐴))
20 fmptsnd.1 . . . . . . . . . . . 12 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
2120eqeq2d 2805 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐴) → (𝐶 = 𝐵𝐶 = 𝐶))
2219, 21anbi12d 630 . . . . . . . . . 10 ((𝜑𝑥 = 𝐴) → ((𝑥 = 𝐴𝐶 = 𝐵) ↔ (𝐴 = 𝐴𝐶 = 𝐶)))
2317, 22sbcied 3743 . . . . . . . . 9 (𝜑 → ([𝐴 / 𝑥](𝑥 = 𝐴𝐶 = 𝐵) ↔ (𝐴 = 𝐴𝐶 = 𝐶)))
2416, 23bitrd 280 . . . . . . . 8 (𝜑 → ([𝐴 / 𝑥][𝐶 / 𝑦](𝑥 = 𝐴𝑦 = 𝐵) ↔ (𝐴 = 𝐴𝐶 = 𝐶)))
256, 7, 24mpbir2and 709 . . . . . . 7 (𝜑[𝐴 / 𝑥][𝐶 / 𝑦](𝑥 = 𝐴𝑦 = 𝐵))
26 opelopabsb 5307 . . . . . . 7 (⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} ↔ [𝐴 / 𝑥][𝐶 / 𝑦](𝑥 = 𝐴𝑦 = 𝐵))
2725, 26sylibr 235 . . . . . 6 (𝜑 → ⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)})
28 eleq1 2870 . . . . . 6 (𝑝 = ⟨𝐴, 𝐶⟩ → (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} ↔ ⟨𝐴, 𝐶⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
2927, 28syl5ibrcom 248 . . . . 5 (𝜑 → (𝑝 = ⟨𝐴, 𝐶⟩ → 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
305, 29syl5bi 243 . . . 4 (𝜑 → (𝑝 ∈ {⟨𝐴, 𝐶⟩} → 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
31 elopab 5304 . . . . 5 (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} ↔ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴𝑦 = 𝐵)))
32 opeq12 4712 . . . . . . . . . . 11 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
3332adantl 482 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
3433eqeq2d 2805 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝑝 = ⟨𝑥, 𝑦⟩ ↔ 𝑝 = ⟨𝐴, 𝐵⟩))
3520adantrr 713 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝐵 = 𝐶)
3635opeq2d 4717 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐶⟩)
37 opex 5248 . . . . . . . . . . . 12 𝐴, 𝐶⟩ ∈ V
3837snid 4506 . . . . . . . . . . 11 𝐴, 𝐶⟩ ∈ {⟨𝐴, 𝐶⟩}
3936, 38syl6eqel 2891 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐶⟩})
40 eleq1 2870 . . . . . . . . . 10 (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐶⟩}))
4139, 40syl5ibrcom 248 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝑝 = ⟨𝐴, 𝐵⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
4234, 41sylbid 241 . . . . . . . 8 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝑝 = ⟨𝑥, 𝑦⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
4342ex 413 . . . . . . 7 (𝜑 → ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑝 = ⟨𝑥, 𝑦⟩ → 𝑝 ∈ {⟨𝐴, 𝐶⟩})))
4443impcomd 412 . . . . . 6 (𝜑 → ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
4544exlimdvv 1912 . . . . 5 (𝜑 → (∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
4631, 45syl5bi 243 . . . 4 (𝜑 → (𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)} → 𝑝 ∈ {⟨𝐴, 𝐶⟩}))
4730, 46impbid 213 . . 3 (𝜑 → (𝑝 ∈ {⟨𝐴, 𝐶⟩} ↔ 𝑝 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)}))
4847eqrdv 2793 . 2 (𝜑 → {⟨𝐴, 𝐶⟩} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = 𝐴𝑦 = 𝐵)})
49 df-mpt 5042 . . 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 2857 1 (𝜑 → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wex 1761  wcel 2081  [wsbc 3706  {csn 4472  cop 4478  {copab 5024  cmpt 5041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-opab 5025  df-mpt 5042
This theorem is referenced by:  fmptapd  6796  fmptpr  6797  mposn  7654
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