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Theorem xpinpreima2 29927
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 5216 . . . . . 6 (𝐸 × 𝐹) ⊆ (V × V)
2 rabss2 3677 . . . . . 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 793 . . . . . . 7 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → 𝑟 ∈ (V × V))
5 simpll 789 . . . . . . . . 9 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → 𝐴𝐸)
6 simprrl 803 . . . . . . . . 9 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → (1st𝑟) ∈ 𝐴)
75, 6sseldd 3596 . . . . . . . 8 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → (1st𝑟) ∈ 𝐸)
8 simplr 791 . . . . . . . . 9 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → 𝐵𝐹)
9 simprrr 804 . . . . . . . . 9 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → (2nd𝑟) ∈ 𝐵)
108, 9sseldd 3596 . . . . . . . 8 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → (2nd𝑟) ∈ 𝐹)
117, 10jca 554 . . . . . . 7 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → ((1st𝑟) ∈ 𝐸 ∧ (2nd𝑟) ∈ 𝐹))
12 elxp7 7186 . . . . . . 7 (𝑟 ∈ (𝐸 × 𝐹) ↔ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐸 ∧ (2nd𝑟) ∈ 𝐹)))
134, 11, 12sylanbrc 697 . . . . . 6 (((𝐴𝐸𝐵𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵))) → 𝑟 ∈ (𝐸 × 𝐹))
1413rabss3d 29323 . . . . 5 ((𝐴𝐸𝐵𝐹) → {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)} ⊆ {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)})
153, 14eqssd 3612 . . . 4 ((𝐴𝐸𝐵𝐹) → {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)} = {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)})
16 xp2 7188 . . . 4 (𝐴 × 𝐵) = {𝑟 ∈ (V × V) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)}
1715, 16syl6reqr 2673 . . 3 ((𝐴𝐸𝐵𝐹) → (𝐴 × 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)})
18 inrab 3891 . . 3 ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵}) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st𝑟) ∈ 𝐴 ∧ (2nd𝑟) ∈ 𝐵)}
1917, 18syl6eqr 2672 . 2 ((𝐴𝐸𝐵𝐹) → (𝐴 × 𝐵) = ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵}))
20 f1stres 7175 . . . . 5 (1st ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐸
21 ffn 6032 . . . . 5 ((1st ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐸 → (1st ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹))
22 fncnvima2 6325 . . . . 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 6194 . . . . . 6 (𝑟 ∈ (𝐸 × 𝐹) → ((1st ↾ (𝐸 × 𝐹))‘𝑟) = (1st𝑟))
2524eleq1d 2684 . . . . 5 (𝑟 ∈ (𝐸 × 𝐹) → (((1st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴 ↔ (1st𝑟) ∈ 𝐴))
2625rabbiia 3180 . . . 4 {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴} = {𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴}
2723, 26eqtri 2642 . . 3 ((1st ↾ (𝐸 × 𝐹)) “ 𝐴) = {𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴}
28 f2ndres 7176 . . . . 5 (2nd ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐹
29 ffn 6032 . . . . 5 ((2nd ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐹 → (2nd ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹))
30 fncnvima2 6325 . . . . 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 6194 . . . . . 6 (𝑟 ∈ (𝐸 × 𝐹) → ((2nd ↾ (𝐸 × 𝐹))‘𝑟) = (2nd𝑟))
3332eleq1d 2684 . . . . 5 (𝑟 ∈ (𝐸 × 𝐹) → (((2nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵 ↔ (2nd𝑟) ∈ 𝐵))
3433rabbiia 3180 . . . 4 {𝑟 ∈ (𝐸 × 𝐹) ∣ ((2nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵} = {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵}
3531, 34eqtri 2642 . . 3 ((2nd ↾ (𝐸 × 𝐹)) “ 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵}
3627, 35ineq12i 3804 . 2 (((1st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ ((2nd ↾ (𝐸 × 𝐹)) “ 𝐵)) = ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1st𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd𝑟) ∈ 𝐵})
3719, 36syl6eqr 2672 1 ((𝐴𝐸𝐵𝐹) → (𝐴 × 𝐵) = (((1st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ ((2nd ↾ (𝐸 × 𝐹)) “ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  {crab 2913  Vcvv 3195  cin 3566  wss 3567   × cxp 5102  ccnv 5103  cres 5106  cima 5107   Fn wfn 5871  wf 5872  cfv 5876  1st c1st 7151  2nd c2nd 7152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-1st 7153  df-2nd 7154
This theorem is referenced by:  cnre2csqima  29931  sxbrsigalem2  30322  sxbrsiga  30326
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