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Theorem xpinpreima2 33868
Description: Rewrite the cartesian product of two sets as the intersection of their preimage by 1st and 2nd, the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.)
Assertion
Ref Expression
xpinpreima2 ((𝐴𝐸𝐵𝐹) → (𝐴 × 𝐵) = (((1st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ ((2nd ↾ (𝐸 × 𝐹)) “ 𝐵)))

Proof of Theorem xpinpreima2
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 xp2 8050 . . . 4 (𝐴 × 𝐵) = {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)}
2 xpss 5705 . . . . . 6 (𝐸 × 𝐹) ⊆ (V × V)
3 rabss2 4088 . . . . . 6 ((𝐸 × 𝐹) ⊆ (V × V) → {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)} ⊆ {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)})
42, 3mp1i 13 . . . . 5 ((𝐴𝐸𝐵𝐹) → {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)} ⊆ {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)})
5 simprl 771 . . . . . . 7 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → 𝑟 ∈ (V × V))
6 simpll 767 . . . . . . . . 9 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → 𝐴𝐸)
7 simprrl 781 . . . . . . . . 9 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → (1st𝑟) ∈ 𝐴)
86, 7sseldd 3996 . . . . . . . 8 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → (1st𝑟) ∈ 𝐸)
9 simplr 769 . . . . . . . . 9 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → 𝐵𝐹)
10 simprrr 782 . . . . . . . . 9 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → (2nd𝑟) ∈ 𝐵)
119, 10sseldd 3996 . . . . . . . 8 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → (2nd𝑟) ∈ 𝐹)
128, 11jca 511 . . . . . . 7 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → ((1st𝑟) ∈ 𝐸 ∧ (2nd𝑟) ∈ 𝐹))
13 elxp7 8048 . . . . . . 7 (𝑟 ∈ (𝐸 × 𝐹) ↔ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐸 ∧ (2nd𝑟) ∈ 𝐹)))
145, 12, 13sylanbrc 583 . . . . . 6 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → 𝑟 ∈ (𝐸 × 𝐹))
1514rabss3d 4091 . . . . 5 ((𝐴𝐸𝐵𝐹) → {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)} ⊆ {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)})
164, 15eqssd 4013 . . . 4 ((𝐴𝐸𝐵𝐹) → {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)} = {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)})
171, 16eqtr4id 2794 . . 3 ((𝐴𝐸𝐵𝐹) → (𝐴 × 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)})
18 inrab 4322 . . 3 ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵}) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)}
1917, 18eqtr4di 2793 . 2 ((𝐴𝐸𝐵𝐹) → (𝐴 × 𝐵) = ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵}))
20 f1stres 8037 . . . . 5 (1st ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐸
21 ffn 6737 . . . . 5 ((1st ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐸 → (1st ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹))
22 fncnvima2 7081 . . . . 5 ((1st ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹) → ((1st ↾ (𝐸 × 𝐹)) “ 𝐴) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴})
2320, 21, 22mp2b 10 . . . 4 ((1st ↾ (𝐸 × 𝐹)) “ 𝐴) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴}
24 fvres 6926 . . . . . 6 (𝑟 ∈ (𝐸 × 𝐹) → ((1st ↾ (𝐸 × 𝐹))‘𝑟) = (1st𝑟))
2524eleq1d 2824 . . . . 5 (𝑟 ∈ (𝐸 × 𝐹) → (((1st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴 ↔ (1st𝑟) ∈ 𝐴))
2625rabbiia 3437 . . . 4 {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴} = {𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴}
2723, 26eqtri 2763 . . 3 ((1st ↾ (𝐸 × 𝐹)) “ 𝐴) = {𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴}
28 f2ndres 8038 . . . . 5 (2nd ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐹
29 ffn 6737 . . . . 5 ((2nd ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐹 → (2nd ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹))
30 fncnvima2 7081 . . . . 5 ((2nd ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹) → ((2nd ↾ (𝐸 × 𝐹)) “ 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((2nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵})
3128, 29, 30mp2b 10 . . . 4 ((2nd ↾ (𝐸 × 𝐹)) “ 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((2nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵}
32 fvres 6926 . . . . . 6 (𝑟 ∈ (𝐸 × 𝐹) → ((2nd ↾ (𝐸 × 𝐹))‘𝑟) = (2nd𝑟))
3332eleq1d 2824 . . . . 5 (𝑟 ∈ (𝐸 × 𝐹) → (((2nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵 ↔ (2nd𝑟) ∈ 𝐵))
3433rabbiia 3437 . . . 4 {𝑟 ∈ (𝐸 × 𝐹) ∣ ((2nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵} = {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵}
3531, 34eqtri 2763 . . 3 ((2nd ↾ (𝐸 × 𝐹)) “ 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵}
3627, 35ineq12i 4226 . 2 (((1st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ ((2nd ↾ (𝐸 × 𝐹)) “ 𝐵)) = ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵})
3719, 36eqtr4di 2793 1 ((𝐴𝐸𝐵𝐹) → (𝐴 × 𝐵) = (((1st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ ((2nd ↾ (𝐸 × 𝐹)) “ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  cin 3962  wss 3963   × cxp 5687  ccnv 5688  cres 5691  cima 5692   Fn wfn 6558  wf 6559  cfv 6563  1st c1st 8011  2nd c2nd 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-1st 8013  df-2nd 8014
This theorem is referenced by:  cnre2csqima  33872  sxbrsigalem2  34268  sxbrsiga  34272
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