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Theorem xpinpreima2 32887
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 8012 . . . 4 (𝐴 × 𝐵) = {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)}
2 xpss 5693 . . . . . 6 (𝐸 × 𝐹) ⊆ (V × V)
3 rabss2 4076 . . . . . 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 770 . . . . . . 7 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → 𝑟 ∈ (V × V))
6 simpll 766 . . . . . . . . 9 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → 𝐴𝐸)
7 simprrl 780 . . . . . . . . 9 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → (1st𝑟) ∈ 𝐴)
86, 7sseldd 3984 . . . . . . . 8 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → (1st𝑟) ∈ 𝐸)
9 simplr 768 . . . . . . . . 9 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → 𝐵𝐹)
10 simprrr 781 . . . . . . . . 9 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → (2nd𝑟) ∈ 𝐵)
119, 10sseldd 3984 . . . . . . . 8 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → (2nd𝑟) ∈ 𝐹)
128, 11jca 513 . . . . . . 7 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → ((1st𝑟) ∈ 𝐸 ∧ (2nd𝑟) ∈ 𝐹))
13 elxp7 8010 . . . . . . 7 (𝑟 ∈ (𝐸 × 𝐹) ↔ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐸 ∧ (2nd𝑟) ∈ 𝐹)))
145, 12, 13sylanbrc 584 . . . . . 6 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → 𝑟 ∈ (𝐸 × 𝐹))
1514rabss3d 4080 . . . . 5 ((𝐴𝐸𝐵𝐹) → {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)} ⊆ {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)})
164, 15eqssd 4000 . . . 4 ((𝐴𝐸𝐵𝐹) → {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)} = {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)})
171, 16eqtr4id 2792 . . 3 ((𝐴𝐸𝐵𝐹) → (𝐴 × 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)})
18 inrab 4307 . . 3 ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵}) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)}
1917, 18eqtr4di 2791 . 2 ((𝐴𝐸𝐵𝐹) → (𝐴 × 𝐵) = ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵}))
20 f1stres 7999 . . . . 5 (1st ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐸
21 ffn 6718 . . . . 5 ((1st ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐸 → (1st ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹))
22 fncnvima2 7063 . . . . 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 6911 . . . . . 6 (𝑟 ∈ (𝐸 × 𝐹) → ((1st ↾ (𝐸 × 𝐹))‘𝑟) = (1st𝑟))
2524eleq1d 2819 . . . . 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 2761 . . 3 ((1st ↾ (𝐸 × 𝐹)) “ 𝐴) = {𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴}
28 f2ndres 8000 . . . . 5 (2nd ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐹
29 ffn 6718 . . . . 5 ((2nd ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐹 → (2nd ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹))
30 fncnvima2 7063 . . . . 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 6911 . . . . . 6 (𝑟 ∈ (𝐸 × 𝐹) → ((2nd ↾ (𝐸 × 𝐹))‘𝑟) = (2nd𝑟))
3332eleq1d 2819 . . . . 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 2761 . . 3 ((2nd ↾ (𝐸 × 𝐹)) “ 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵}
3627, 35ineq12i 4211 . 2 (((1st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ ((2nd ↾ (𝐸 × 𝐹)) “ 𝐵)) = ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵})
3719, 36eqtr4di 2791 1 ((𝐴𝐸𝐵𝐹) → (𝐴 × 𝐵) = (((1st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ ((2nd ↾ (𝐸 × 𝐹)) “ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {crab 3433  Vcvv 3475  cin 3948  wss 3949   × cxp 5675  ccnv 5676  cres 5679  cima 5680   Fn wfn 6539  wf 6540  cfv 6544  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-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-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-1st 7975  df-2nd 7976
This theorem is referenced by:  cnre2csqima  32891  sxbrsigalem2  33285  sxbrsiga  33289
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