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Theorem mpt2sn 7388
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 6521 . . . 4 ((𝐴𝑉𝐵𝑊) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
213adant3 1124 . . 3 ((𝐴𝑉𝐵𝑊𝐸𝑈) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩})
32mpteq1d 4846 . 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 7354 . . . 4 (𝑥 ∈ {𝐴}, 𝑦 ∈ {𝐵} ↦ 𝐶) = (𝑝 ∈ ({𝐴} × {𝐵}) ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶)
64, 5eqtri 2746 . . 3 𝐹 = (𝑝 ∈ ({𝐴} × {𝐵}) ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶)
76a1i 11 . 2 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐹 = (𝑝 ∈ ({𝐴} × {𝐵}) ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶))
8 op2ndg 7298 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
9 fveq2 6304 . . . . . . . . 9 (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd𝑝) = (2nd ‘⟨𝐴, 𝐵⟩))
109eqcomd 2730 . . . . . . . 8 (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd ‘⟨𝐴, 𝐵⟩) = (2nd𝑝))
1110eqeq1d 2726 . . . . . . 7 (𝑝 = ⟨𝐴, 𝐵⟩ → ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐵 ↔ (2nd𝑝) = 𝐵))
128, 11syl5ibcom 235 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd𝑝) = 𝐵))
13123adant3 1124 . . . . 5 ((𝐴𝑉𝐵𝑊𝐸𝑈) → (𝑝 = ⟨𝐴, 𝐵⟩ → (2nd𝑝) = 𝐵))
1413imp 444 . . . 4 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (2nd𝑝) = 𝐵)
15 op1stg 7297 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
16 fveq2 6304 . . . . . . . . 9 (𝑝 = ⟨𝐴, 𝐵⟩ → (1st𝑝) = (1st ‘⟨𝐴, 𝐵⟩))
1716eqcomd 2730 . . . . . . . 8 (𝑝 = ⟨𝐴, 𝐵⟩ → (1st ‘⟨𝐴, 𝐵⟩) = (1st𝑝))
1817eqeq1d 2726 . . . . . . 7 (𝑝 = ⟨𝐴, 𝐵⟩ → ((1st ‘⟨𝐴, 𝐵⟩) = 𝐴 ↔ (1st𝑝) = 𝐴))
1915, 18syl5ibcom 235 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (𝑝 = ⟨𝐴, 𝐵⟩ → (1st𝑝) = 𝐴))
20193adant3 1124 . . . . 5 ((𝐴𝑉𝐵𝑊𝐸𝑈) → (𝑝 = ⟨𝐴, 𝐵⟩ → (1st𝑝) = 𝐴))
2120imp 444 . . . 4 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (1st𝑝) = 𝐴)
22 simp1 1128 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐴𝑉)
23 simpl2 1206 . . . . . . . 8 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑥 = 𝐴) → 𝐵𝑊)
24 mpt2sn.a . . . . . . . . . 10 (𝑥 = 𝐴𝐶 = 𝐷)
2524adantl 473 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷)
26 mpt2sn.b . . . . . . . . 9 (𝑦 = 𝐵𝐷 = 𝐸)
2725, 26sylan9eq 2778 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐸)
2823, 27csbied 3666 . . . . . . 7 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑥 = 𝐴) → 𝐵 / 𝑦𝐶 = 𝐸)
2922, 28csbied 3666 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐸)
3029adantr 472 . . . . 5 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐸)
31 csbeq1 3642 . . . . . . . 8 ((1st𝑝) = 𝐴(1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶)
3231eqeq1d 2726 . . . . . . 7 ((1st𝑝) = 𝐴 → ((1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸))
3332adantl 473 . . . . . 6 (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → ((1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸))
34 csbeq1 3642 . . . . . . . . 9 ((2nd𝑝) = 𝐵(2nd𝑝) / 𝑦𝐶 = 𝐵 / 𝑦𝐶)
3534adantr 472 . . . . . . . 8 (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → (2nd𝑝) / 𝑦𝐶 = 𝐵 / 𝑦𝐶)
3635csbeq2dv 4100 . . . . . . 7 (((2nd𝑝) = 𝐵 ∧ (1st𝑝) = 𝐴) → 𝐴 / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
3736eqeq1d 2726 . . . . . 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 717 . . 3 (((𝐴𝑉𝐵𝑊𝐸𝑈) ∧ 𝑝 = ⟨𝐴, 𝐵⟩) → (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶 = 𝐸)
41 opex 5037 . . . 4 𝐴, 𝐵⟩ ∈ V
4241a1i 11 . . 3 ((𝐴𝑉𝐵𝑊𝐸𝑈) → ⟨𝐴, 𝐵⟩ ∈ V)
43 simp3 1130 . . 3 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐸𝑈)
4440, 42, 43fmptsnd 6551 . 2 ((𝐴𝑉𝐵𝑊𝐸𝑈) → {⟨⟨𝐴, 𝐵⟩, 𝐸⟩} = (𝑝 ∈ {⟨𝐴, 𝐵⟩} ↦ (1st𝑝) / 𝑥(2nd𝑝) / 𝑦𝐶))
453, 7, 443eqtr4d 2768 1 ((𝐴𝑉𝐵𝑊𝐸𝑈) → 𝐹 = {⟨⟨𝐴, 𝐵⟩, 𝐸⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1596  wcel 2103  Vcvv 3304  csb 3639  {csn 4285  cop 4291  cmpt 4837   × cxp 5216  cfv 6001  cmpt2 6767  1st c1st 7283  2nd c2nd 7284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-oprab 6769  df-mpt2 6770  df-1st 7285  df-2nd 7286
This theorem is referenced by:  mat1dim0  20402  mat1dimid  20403  mat1dimmul  20405  d1mat2pmat  20667
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