MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpt2sn Structured version   Visualization version   GIF version

Theorem mpt2sn 7417
Description: An operation (in maps-to notation) on two singletons. (Contributed by AV, 4-Aug-2019.)
Hypotheses
Ref Expression
mpt2sn.f 𝐹 = (𝑥 ∈ {𝐴}, 𝑦 ∈ {𝐵} ↦ 𝐶)
mpt2sn.a (𝑥 = 𝐴𝐶 = 𝐷)
mpt2sn.b (𝑦 = 𝐵𝐷 = 𝐸)
Assertion
Ref Expression
mpt2sn ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐹 = {⟨⟨𝐴, 𝐵⟩, 𝐸⟩})
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐸,𝑦   𝑥,𝑈,𝑦   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem mpt2sn
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 xpsng 6547 . . . 4 ((𝐴𝑉𝐵𝑊) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
213adant3 1126 . . 3 ((𝐴𝑉𝐵𝑊𝐸𝑈) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
32mpteq1d 4872 . 2 ((𝐴𝑉𝐵𝑊𝐸𝑈) → (𝑝 ∈ ({𝐴} × {𝐵}) ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶) = (𝑝 ∈ {⟨𝐴, 𝐵⟩} ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶))
4 mpt2sn.f . . . 4 𝐹 = (𝑥 ∈ {𝐴}, 𝑦 ∈ {𝐵} ↦ 𝐶)
5 mpt2mpts 7382 . . . 4 (𝑥 ∈ {𝐴}, 𝑦 ∈ {𝐵} ↦ 𝐶) = (𝑝 ∈ ({𝐴} × {𝐵}) ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶)
64, 5eqtri 2793 . . 3 𝐹 = (𝑝 ∈ ({𝐴} × {𝐵}) ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶)
76a1i 11 . 2 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐹 = (𝑝 ∈ ({𝐴} × {𝐵}) ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶))
8 op2ndg 7326 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
9 fveq2 6330 . . . . . . . . 9 (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd𝑝) = (2nd ‘⟨𝐴, 𝐵⟩))
109eqcomd 2777 . . . . . . . 8 (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd ‘⟨𝐴, 𝐵⟩) = (2nd𝑝))
1110eqeq1d 2773 . . . . . . 7 (𝑝 = ⟨𝐴, 𝐵⟩ → ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐵 ↔ (2nd𝑝) = 𝐵))
128, 11syl5ibcom 235 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd𝑝) = 𝐵))
13123adant3 1126 . . . . 5 ((𝐴𝑉𝐵𝑊𝐸𝑈) → (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd𝑝) = 𝐵))
1413imp 393 . . . 4 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (2nd𝑝) = 𝐵)
15 op1stg 7325 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
16 fveq2 6330 . . . . . . . . 9 (𝑝 = ⟨𝐴, 𝐵⟩ → (1st𝑝) = (1st ‘⟨𝐴, 𝐵⟩))
1716eqcomd 2777 . . . . . . . 8 (𝑝 = ⟨𝐴, 𝐵⟩ → (1st ‘⟨𝐴, 𝐵⟩) = (1st𝑝))
1817eqeq1d 2773 . . . . . . 7 (𝑝 = ⟨𝐴, 𝐵⟩ → ((1st ‘⟨𝐴, 𝐵⟩) = 𝐴 ↔ (1st𝑝) = 𝐴))
1915, 18syl5ibcom 235 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (𝑝 = ⟨𝐴, 𝐵⟩ → (1st𝑝) = 𝐴))
20193adant3 1126 . . . . 5 ((𝐴𝑉𝐵𝑊𝐸𝑈) → (𝑝 = ⟨𝐴, 𝐵⟩ → (1st𝑝) = 𝐴))
2120imp 393 . . . 4 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (1st𝑝) = 𝐴)
22 simp1 1130 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐴𝑉)
23 simpl2 1229 . . . . . . . 8 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑥 = 𝐴) → 𝐵𝑊)
24 mpt2sn.a . . . . . . . . . 10 (𝑥 = 𝐴𝐶 = 𝐷)
2524adantl 467 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷)
26 mpt2sn.b . . . . . . . . 9 (𝑦 = 𝐵𝐷 = 𝐸)
2725, 26sylan9eq 2825 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐸)
2823, 27csbied 3709 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑥 = 𝐴) → 𝐵 / 𝑦𝐶 = 𝐸)
2922, 28csbied 3709 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐸)
3029adantr 466 . . . . 5 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐸)
31 csbeq1 3685 . . . . . . . 8 ((1st𝑝) = 𝐴(1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶)
3231eqeq1d 2773 . . . . . . 7 ((1st𝑝) = 𝐴 → ((1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸))
3332adantl 467 . . . . . 6 (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → ((1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸))
34 csbeq1 3685 . . . . . . . . 9 ((2nd𝑝) = 𝐵(2nd𝑝) / 𝑦𝐶 = 𝐵 / 𝑦𝐶)
3534adantr 466 . . . . . . . 8 (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → (2nd𝑝) / 𝑦𝐶 = 𝐵 / 𝑦𝐶)
3635csbeq2dv 4136 . . . . . . 7 (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → 𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
3736eqeq1d 2773 . . . . . 6 (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → (𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐸))
3833, 37bitrd 268 . . . . 5 (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → ((1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐸))
3930, 38syl5ibrcom 237 . . . 4 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸))
4014, 21, 39mp2and 679 . . 3 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸)
41 opex 5060 . . . 4 𝐴, 𝐵⟩ ∈ V
4241a1i 11 . . 3 ((𝐴𝑉𝐵𝑊𝐸𝑈) → ⟨𝐴, 𝐵⟩ ∈ V)
43 simp3 1132 . . 3 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐸𝑈)
4440, 42, 43fmptsnd 6577 . 2 ((𝐴𝑉𝐵𝑊𝐸𝑈) → {⟨⟨𝐴, 𝐵⟩, 𝐸⟩} = (𝑝 ∈ {⟨𝐴, 𝐵⟩} ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶))
453, 7, 443eqtr4d 2815 1 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐹 = {⟨⟨𝐴, 𝐵⟩, 𝐸⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  Vcvv 3351  csb 3682  {csn 4316  cop 4322  cmpt 4863   × cxp 5247  cfv 6029  cmpt2 6793  1st c1st 7311  2nd c2nd 7312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-oprab 6795  df-mpt2 6796  df-1st 7313  df-2nd 7314
This theorem is referenced by:  mat1dim0  20490  mat1dimid  20491  mat1dimmul  20493  d1mat2pmat  20757
  Copyright terms: Public domain W3C validator