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Theorem mposn 8089
Description: An operation (in maps-to notation) on two singletons. (Contributed by AV, 4-Aug-2019.)
Hypotheses
Ref Expression
mposn.f 𝐹 = (𝑥 ∈ {𝐴}, 𝑦 ∈ {𝐵} ↦ 𝐶)
mposn.a (𝑥 = 𝐴𝐶 = 𝐷)
mposn.b (𝑦 = 𝐵𝐷 = 𝐸)
Assertion
Ref Expression
mposn ((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 mposn
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 xpsng 7137 . . . 4 ((𝐴𝑉𝐵𝑊) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
213adant3 1133 . . 3 ((𝐴𝑉𝐵𝑊𝐸𝑈) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
32mpteq1d 5244 . 2 ((𝐴𝑉𝐵𝑊𝐸𝑈) → (𝑝 ∈ ({𝐴} × {𝐵}) ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶) = (𝑝 ∈ {⟨𝐴, 𝐵⟩} ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶))
4 mposn.f . . . 4 𝐹 = (𝑥 ∈ {𝐴}, 𝑦 ∈ {𝐵} ↦ 𝐶)
5 mpompts 8051 . . . 4 (𝑥 ∈ {𝐴}, 𝑦 ∈ {𝐵} ↦ 𝐶) = (𝑝 ∈ ({𝐴} × {𝐵}) ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶)
64, 5eqtri 2761 . . 3 𝐹 = (𝑝 ∈ ({𝐴} × {𝐵}) ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶)
76a1i 11 . 2 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐹 = (𝑝 ∈ ({𝐴} × {𝐵}) ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶))
8 op2ndg 7988 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
9 fveq2 6892 . . . . . . . . 9 (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd𝑝) = (2nd ‘⟨𝐴, 𝐵⟩))
109eqcomd 2739 . . . . . . . 8 (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd ‘⟨𝐴, 𝐵⟩) = (2nd𝑝))
1110eqeq1d 2735 . . . . . . 7 (𝑝 = ⟨𝐴, 𝐵⟩ → ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐵 ↔ (2nd𝑝) = 𝐵))
128, 11syl5ibcom 244 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd𝑝) = 𝐵))
13123adant3 1133 . . . . 5 ((𝐴𝑉𝐵𝑊𝐸𝑈) → (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd𝑝) = 𝐵))
1413imp 408 . . . 4 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (2nd𝑝) = 𝐵)
15 op1stg 7987 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
16 fveq2 6892 . . . . . . . . 9 (𝑝 = ⟨𝐴, 𝐵⟩ → (1st𝑝) = (1st ‘⟨𝐴, 𝐵⟩))
1716eqcomd 2739 . . . . . . . 8 (𝑝 = ⟨𝐴, 𝐵⟩ → (1st ‘⟨𝐴, 𝐵⟩) = (1st𝑝))
1817eqeq1d 2735 . . . . . . 7 (𝑝 = ⟨𝐴, 𝐵⟩ → ((1st ‘⟨𝐴, 𝐵⟩) = 𝐴 ↔ (1st𝑝) = 𝐴))
1915, 18syl5ibcom 244 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (𝑝 = ⟨𝐴, 𝐵⟩ → (1st𝑝) = 𝐴))
20193adant3 1133 . . . . 5 ((𝐴𝑉𝐵𝑊𝐸𝑈) → (𝑝 = ⟨𝐴, 𝐵⟩ → (1st𝑝) = 𝐴))
2120imp 408 . . . 4 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (1st𝑝) = 𝐴)
22 simp1 1137 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐴𝑉)
23 simpl2 1193 . . . . . . . 8 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑥 = 𝐴) → 𝐵𝑊)
24 mposn.a . . . . . . . . . 10 (𝑥 = 𝐴𝐶 = 𝐷)
2524adantl 483 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷)
26 mposn.b . . . . . . . . 9 (𝑦 = 𝐵𝐷 = 𝐸)
2725, 26sylan9eq 2793 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐸)
2823, 27csbied 3932 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑥 = 𝐴) → 𝐵 / 𝑦𝐶 = 𝐸)
2922, 28csbied 3932 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐸)
3029adantr 482 . . . . 5 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐸)
31 csbeq1 3897 . . . . . . . 8 ((1st𝑝) = 𝐴(1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶)
3231eqeq1d 2735 . . . . . . 7 ((1st𝑝) = 𝐴 → ((1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸))
3332adantl 483 . . . . . 6 (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → ((1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸))
34 csbeq1 3897 . . . . . . . . 9 ((2nd𝑝) = 𝐵(2nd𝑝) / 𝑦𝐶 = 𝐵 / 𝑦𝐶)
3534adantr 482 . . . . . . . 8 (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → (2nd𝑝) / 𝑦𝐶 = 𝐵 / 𝑦𝐶)
3635csbeq2dv 3901 . . . . . . 7 (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → 𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
3736eqeq1d 2735 . . . . . 6 (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → (𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐸))
3833, 37bitrd 279 . . . . 5 (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → ((1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐸))
3930, 38syl5ibrcom 246 . . . 4 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸))
4014, 21, 39mp2and 698 . . 3 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸)
41 opex 5465 . . . 4 𝐴, 𝐵⟩ ∈ V
4241a1i 11 . . 3 ((𝐴𝑉𝐵𝑊𝐸𝑈) → ⟨𝐴, 𝐵⟩ ∈ V)
43 simp3 1139 . . 3 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐸𝑈)
4440, 42, 43fmptsnd 7167 . 2 ((𝐴𝑉𝐵𝑊𝐸𝑈) → {⟨⟨𝐴, 𝐵⟩, 𝐸⟩} = (𝑝 ∈ {⟨𝐴, 𝐵⟩} ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶))
453, 7, 443eqtr4d 2783 1 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐹 = {⟨⟨𝐴, 𝐵⟩, 𝐸⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3475  csb 3894  {csn 4629  cop 4635  cmpt 5232   × cxp 5675  cfv 6544  cmpo 7411  1st c1st 7973  2nd c2nd 7974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976
This theorem is referenced by:  mat1dim0  21975  mat1dimid  21976  mat1dimmul  21978  d1mat2pmat  22241
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