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Theorem mpt2sn 7216
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 6363 . . . 4 ((𝐴𝑉𝐵𝑊) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
213adant3 1079 . . 3 ((𝐴𝑉𝐵𝑊𝐸𝑈) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
32mpteq1d 4700 . 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 7182 . . . 4 (𝑥 ∈ {𝐴}, 𝑦 ∈ {𝐵} ↦ 𝐶) = (𝑝 ∈ ({𝐴} × {𝐵}) ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶)
64, 5eqtri 2643 . . 3 𝐹 = (𝑝 ∈ ({𝐴} × {𝐵}) ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶)
76a1i 11 . 2 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐹 = (𝑝 ∈ ({𝐴} × {𝐵}) ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶))
8 op2ndg 7129 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
9 fveq2 6150 . . . . . . . . 9 (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd𝑝) = (2nd ‘⟨𝐴, 𝐵⟩))
109eqcomd 2627 . . . . . . . 8 (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd ‘⟨𝐴, 𝐵⟩) = (2nd𝑝))
1110eqeq1d 2623 . . . . . . 7 (𝑝 = ⟨𝐴, 𝐵⟩ → ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐵 ↔ (2nd𝑝) = 𝐵))
128, 11syl5ibcom 235 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd𝑝) = 𝐵))
13123adant3 1079 . . . . 5 ((𝐴𝑉𝐵𝑊𝐸𝑈) → (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd𝑝) = 𝐵))
1413imp 445 . . . 4 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (2nd𝑝) = 𝐵)
15 op1stg 7128 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
16 fveq2 6150 . . . . . . . . 9 (𝑝 = ⟨𝐴, 𝐵⟩ → (1st𝑝) = (1st ‘⟨𝐴, 𝐵⟩))
1716eqcomd 2627 . . . . . . . 8 (𝑝 = ⟨𝐴, 𝐵⟩ → (1st ‘⟨𝐴, 𝐵⟩) = (1st𝑝))
1817eqeq1d 2623 . . . . . . 7 (𝑝 = ⟨𝐴, 𝐵⟩ → ((1st ‘⟨𝐴, 𝐵⟩) = 𝐴 ↔ (1st𝑝) = 𝐴))
1915, 18syl5ibcom 235 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (𝑝 = ⟨𝐴, 𝐵⟩ → (1st𝑝) = 𝐴))
20193adant3 1079 . . . . 5 ((𝐴𝑉𝐵𝑊𝐸𝑈) → (𝑝 = ⟨𝐴, 𝐵⟩ → (1st𝑝) = 𝐴))
2120imp 445 . . . 4 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (1st𝑝) = 𝐴)
22 simp1 1059 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐴𝑉)
23 simpl2 1063 . . . . . . . 8 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑥 = 𝐴) → 𝐵𝑊)
24 mpt2sn.a . . . . . . . . . 10 (𝑥 = 𝐴𝐶 = 𝐷)
2524adantl 482 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷)
26 mpt2sn.b . . . . . . . . 9 (𝑦 = 𝐵𝐷 = 𝐸)
2725, 26sylan9eq 2675 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐸)
2823, 27csbied 3542 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑥 = 𝐴) → 𝐵 / 𝑦𝐶 = 𝐸)
2922, 28csbied 3542 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐸)
3029adantr 481 . . . . 5 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐸)
31 csbeq1 3518 . . . . . . . 8 ((1st𝑝) = 𝐴(1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶)
3231eqeq1d 2623 . . . . . . 7 ((1st𝑝) = 𝐴 → ((1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸))
3332adantl 482 . . . . . 6 (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → ((1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸))
34 csbeq1 3518 . . . . . . . . 9 ((2nd𝑝) = 𝐵(2nd𝑝) / 𝑦𝐶 = 𝐵 / 𝑦𝐶)
3534adantr 481 . . . . . . . 8 (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → (2nd𝑝) / 𝑦𝐶 = 𝐵 / 𝑦𝐶)
3635csbeq2dv 3966 . . . . . . 7 (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → 𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
3736eqeq1d 2623 . . . . . 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 714 . . 3 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸)
41 opex 4895 . . . 4 𝐴, 𝐵⟩ ∈ V
4241a1i 11 . . 3 ((𝐴𝑉𝐵𝑊𝐸𝑈) → ⟨𝐴, 𝐵⟩ ∈ V)
43 simp3 1061 . . 3 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐸𝑈)
4440, 42, 43fmptsnd 6392 . 2 ((𝐴𝑉𝐵𝑊𝐸𝑈) → {⟨⟨𝐴, 𝐵⟩, 𝐸⟩} = (𝑝 ∈ {⟨𝐴, 𝐵⟩} ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶))
453, 7, 443eqtr4d 2665 1 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐹 = {⟨⟨𝐴, 𝐵⟩, 𝐸⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3186  csb 3515  {csn 4150  cop 4156  cmpt 4675   × cxp 5074  cfv 5849  cmpt2 6609  1st c1st 7114  2nd c2nd 7115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-oprab 6611  df-mpt2 6612  df-1st 7116  df-2nd 7117
This theorem is referenced by:  mat1dim0  20201  mat1dimid  20202  mat1dimmul  20204  d1mat2pmat  20466
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