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Theorem xpinpreima2 30081
 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 5159 . . . . . 6 (𝐸 × 𝐹) ⊆ (V × V)
2 rabss2 3718 . . . . . 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 809 . . . . . . 7 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → 𝑟 ∈ (V × V))
5 simpll 805 . . . . . . . . 9 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → 𝐴𝐸)
6 simprrl 821 . . . . . . . . 9 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → (1st𝑟) ∈ 𝐴)
75, 6sseldd 3637 . . . . . . . 8 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → (1st𝑟) ∈ 𝐸)
8 simplr 807 . . . . . . . . 9 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → 𝐵𝐹)
9 simprrr 822 . . . . . . . . 9 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → (2nd𝑟) ∈ 𝐵)
108, 9sseldd 3637 . . . . . . . 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 7245 . . . . . . 7 (𝑟 ∈ (𝐸 × 𝐹) ↔ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐸 ∧ (2nd𝑟) ∈ 𝐹)))
134, 11, 12sylanbrc 699 . . . . . 6 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → 𝑟 ∈ (𝐸 × 𝐹))
1413rabss3d 29477 . . . . 5 ((𝐴𝐸𝐵𝐹) → {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)} ⊆ {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)})
153, 14eqssd 3653 . . . 4 ((𝐴𝐸𝐵𝐹) → {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)} = {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)})
16 xp2 7247 . . . 4 (𝐴 × 𝐵) = {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)}
1715, 16syl6reqr 2704 . . 3 ((𝐴𝐸𝐵𝐹) → (𝐴 × 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)})
18 inrab 3932 . . 3 ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵}) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)}
1917, 18syl6eqr 2703 . 2 ((𝐴𝐸𝐵𝐹) → (𝐴 × 𝐵) = ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵}))
20 f1stres 7234 . . . . 5 (1st ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐸
21 ffn 6083 . . . . 5 ((1st ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐸 → (1st ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹))
22 fncnvima2 6379 . . . . 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 6245 . . . . . 6 (𝑟 ∈ (𝐸 × 𝐹) → ((1st ↾ (𝐸 × 𝐹))‘𝑟) = (1st𝑟))
2524eleq1d 2715 . . . . 5 (𝑟 ∈ (𝐸 × 𝐹) → (((1st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴 ↔ (1st𝑟) ∈ 𝐴))
2625rabbiia 3215 . . . 4 {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴} = {𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴}
2723, 26eqtri 2673 . . 3 ((1st ↾ (𝐸 × 𝐹)) “ 𝐴) = {𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴}
28 f2ndres 7235 . . . . 5 (2nd ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐹
29 ffn 6083 . . . . 5 ((2nd ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐹 → (2nd ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹))
30 fncnvima2 6379 . . . . 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 6245 . . . . . 6 (𝑟 ∈ (𝐸 × 𝐹) → ((2nd ↾ (𝐸 × 𝐹))‘𝑟) = (2nd𝑟))
3332eleq1d 2715 . . . . 5 (𝑟 ∈ (𝐸 × 𝐹) → (((2nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵 ↔ (2nd𝑟) ∈ 𝐵))
3433rabbiia 3215 . . . 4 {𝑟 ∈ (𝐸 × 𝐹) ∣ ((2nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵} = {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵}
3531, 34eqtri 2673 . . 3 ((2nd ↾ (𝐸 × 𝐹)) “ 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵}
3627, 35ineq12i 3845 . 2 (((1st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ ((2nd ↾ (𝐸 × 𝐹)) “ 𝐵)) = ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵})
3719, 36syl6eqr 2703 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 1523   ∈ wcel 2030  {crab 2945  Vcvv 3231   ∩ cin 3606   ⊆ wss 3607   × cxp 5141  ◡ccnv 5142   ↾ cres 5145   “ cima 5146   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  1st c1st 7208  2nd c2nd 7209 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-1st 7210  df-2nd 7211 This theorem is referenced by:  cnre2csqima  30085  sxbrsigalem2  30476  sxbrsiga  30480
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