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Theorem xpinpreima2 29281
 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 xpss 5149 . . . . . 6 (𝐸 × 𝐹) ⊆ (V × V)
2 rabss2 3648 . . . . . 6 ((𝐸 × 𝐹) ⊆ (V × V) → {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)} ⊆ {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)})
31, 2mp1i 13 . . . . 5 ((𝐴𝐸𝐵𝐹) → {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)} ⊆ {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)})
4 simprl 790 . . . . . . 7 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → 𝑟 ∈ (V × V))
5 simpll 786 . . . . . . . . 9 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → 𝐴𝐸)
6 simprrl 800 . . . . . . . . 9 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → (1st𝑟) ∈ 𝐴)
75, 6sseldd 3569 . . . . . . . 8 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → (1st𝑟) ∈ 𝐸)
8 simplr 788 . . . . . . . . 9 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → 𝐵𝐹)
9 simprrr 801 . . . . . . . . 9 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → (2nd𝑟) ∈ 𝐵)
108, 9sseldd 3569 . . . . . . . 8 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → (2nd𝑟) ∈ 𝐹)
117, 10jca 553 . . . . . . 7 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → ((1st𝑟) ∈ 𝐸 ∧ (2nd𝑟) ∈ 𝐹))
12 elxp7 7092 . . . . . . 7 (𝑟 ∈ (𝐸 × 𝐹) ↔ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐸 ∧ (2nd𝑟) ∈ 𝐹)))
134, 11, 12sylanbrc 695 . . . . . 6 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → 𝑟 ∈ (𝐸 × 𝐹))
1413rabss3d 28736 . . . . 5 ((𝐴𝐸𝐵𝐹) → {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)} ⊆ {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)})
153, 14eqssd 3585 . . . 4 ((𝐴𝐸𝐵𝐹) → {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)} = {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)})
16 xp2 7094 . . . 4 (𝐴 × 𝐵) = {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)}
1715, 16syl6reqr 2663 . . 3 ((𝐴𝐸𝐵𝐹) → (𝐴 × 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)})
18 inrab 3858 . . 3 ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵}) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)}
1917, 18syl6eqr 2662 . 2 ((𝐴𝐸𝐵𝐹) → (𝐴 × 𝐵) = ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵}))
20 f1stres 7081 . . . . 5 (1st ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐸
21 ffn 5958 . . . . 5 ((1st ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐸 → (1st ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹))
22 fncnvima2 6247 . . . . 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 6117 . . . . . 6 (𝑟 ∈ (𝐸 × 𝐹) → ((1st ↾ (𝐸 × 𝐹))‘𝑟) = (1st𝑟))
2524eleq1d 2672 . . . . 5 (𝑟 ∈ (𝐸 × 𝐹) → (((1st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴 ↔ (1st𝑟) ∈ 𝐴))
2625rabbiia 3161 . . . 4 {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴} = {𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴}
2723, 26eqtri 2632 . . 3 ((1st ↾ (𝐸 × 𝐹)) “ 𝐴) = {𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴}
28 f2ndres 7082 . . . . 5 (2nd ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐹
29 ffn 5958 . . . . 5 ((2nd ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐹 → (2nd ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹))
30 fncnvima2 6247 . . . . 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 6117 . . . . . 6 (𝑟 ∈ (𝐸 × 𝐹) → ((2nd ↾ (𝐸 × 𝐹))‘𝑟) = (2nd𝑟))
3332eleq1d 2672 . . . . 5 (𝑟 ∈ (𝐸 × 𝐹) → (((2nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵 ↔ (2nd𝑟) ∈ 𝐵))
3433rabbiia 3161 . . . 4 {𝑟 ∈ (𝐸 × 𝐹) ∣ ((2nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵} = {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵}
3531, 34eqtri 2632 . . 3 ((2nd ↾ (𝐸 × 𝐹)) “ 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵}
3627, 35ineq12i 3774 . 2 (((1st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ ((2nd ↾ (𝐸 × 𝐹)) “ 𝐵)) = ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵})
3719, 36syl6eqr 2662 1 ((𝐴𝐸𝐵𝐹) → (𝐴 × 𝐵) = (((1st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ ((2nd ↾ (𝐸 × 𝐹)) “ 𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540   × cxp 5036  ◡ccnv 5037   ↾ cres 5040   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  1st c1st 7057  2nd c2nd 7058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-1st 7059  df-2nd 7060 This theorem is referenced by:  cnre2csqima  29285  sxbrsigalem2  29675  sxbrsiga  29679
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