Theorem xpinpreima2 29953

Description: Rewrite the cartesian product of two sets as the intersection of their preimage by 1^st and 2^nd, the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.)

Assertion

Ref	Expression
xpinpreima2	⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → (𝐴 × 𝐵) = ((◡(1st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ (◡(2nd ↾ (𝐸 × 𝐹)) “ 𝐵)))

Proof of Theorem xpinpreima2

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpss 5226	. . . . . 6 ⊢ (𝐸 × 𝐹) ⊆ (V × V)
2		rabss2 3685	. . . . . 6 ⊢ ((𝐸 × 𝐹) ⊆ (V × V) → {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)} ⊆ {𝑟 ∈ (V × V) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)})
3	1, 2	mp1i 13	. . . . 5 ⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)} ⊆ {𝑟 ∈ (V × V) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)})
4		simprl 794	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵))) → 𝑟 ∈ (V × V))
5		simpll 790	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵))) → 𝐴 ⊆ 𝐸)
6		simprrl 804	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵))) → (1st ‘𝑟) ∈ 𝐴)
7	5, 6	sseldd 3604	. . . . . . . 8 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵))) → (1st ‘𝑟) ∈ 𝐸)
8		simplr 792	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵))) → 𝐵 ⊆ 𝐹)
9		simprrr 805	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵))) → (2nd ‘𝑟) ∈ 𝐵)
10	8, 9	sseldd 3604	. . . . . . . 8 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵))) → (2nd ‘𝑟) ∈ 𝐹)
11	7, 10	jca 554	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵))) → ((1st ‘𝑟) ∈ 𝐸 ∧ (2nd ‘𝑟) ∈ 𝐹))
12		elxp7 7201	. . . . . . 7 ⊢ (𝑟 ∈ (𝐸 × 𝐹) ↔ (𝑟 ∈ (V × V) ∧ ((1st ‘𝑟) ∈ 𝐸 ∧ (2nd ‘𝑟) ∈ 𝐹)))
13	4, 11, 12	sylanbrc 698	. . . . . 6 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵))) → 𝑟 ∈ (𝐸 × 𝐹))
14	13	rabss3d 29351	. . . . 5 ⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → {𝑟 ∈ (V × V) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)} ⊆ {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)})
15	3, 14	eqssd 3620	. . . 4 ⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)} = {𝑟 ∈ (V × V) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)})
16		xp2 7203	. . . 4 ⊢ (𝐴 × 𝐵) = {𝑟 ∈ (V × V) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)}
17	15, 16	syl6reqr 2675	. . 3 ⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → (𝐴 × 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)})
18		inrab 3899	. . 3 ⊢ ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1st ‘𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd ‘𝑟) ∈ 𝐵}) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)}
19	17, 18	syl6eqr 2674	. 2 ⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → (𝐴 × 𝐵) = ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1st ‘𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd ‘𝑟) ∈ 𝐵}))
20		f1stres 7190	. . . . 5 ⊢ (1st ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐸
21		ffn 6045	. . . . 5 ⊢ ((1st ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐸 → (1st ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹))
22		fncnvima2 6339	. . . . 5 ⊢ ((1st ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹) → (◡(1st ↾ (𝐸 × 𝐹)) “ 𝐴) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴})
23	20, 21, 22	mp2b 10	. . . 4 ⊢ (◡(1st ↾ (𝐸 × 𝐹)) “ 𝐴) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴}
24		fvres 6207	. . . . . 6 ⊢ (𝑟 ∈ (𝐸 × 𝐹) → ((1st ↾ (𝐸 × 𝐹))‘𝑟) = (1st ‘𝑟))
25	24	eleq1d 2686	. . . . 5 ⊢ (𝑟 ∈ (𝐸 × 𝐹) → (((1st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴 ↔ (1st ‘𝑟) ∈ 𝐴))
26	25	rabbiia 3185	. . . 4 ⊢ {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴} = {𝑟 ∈ (𝐸 × 𝐹) ∣ (1st ‘𝑟) ∈ 𝐴}
27	23, 26	eqtri 2644	. . 3 ⊢ (◡(1st ↾ (𝐸 × 𝐹)) “ 𝐴) = {𝑟 ∈ (𝐸 × 𝐹) ∣ (1st ‘𝑟) ∈ 𝐴}
28		f2ndres 7191	. . . . 5 ⊢ (2nd ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐹
29		ffn 6045	. . . . 5 ⊢ ((2nd ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐹 → (2nd ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹))
30		fncnvima2 6339	. . . . 5 ⊢ ((2nd ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹) → (◡(2nd ↾ (𝐸 × 𝐹)) “ 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((2nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵})
31	28, 29, 30	mp2b 10	. . . 4 ⊢ (◡(2nd ↾ (𝐸 × 𝐹)) “ 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((2nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵}
32		fvres 6207	. . . . . 6 ⊢ (𝑟 ∈ (𝐸 × 𝐹) → ((2nd ↾ (𝐸 × 𝐹))‘𝑟) = (2nd ‘𝑟))
33	32	eleq1d 2686	. . . . 5 ⊢ (𝑟 ∈ (𝐸 × 𝐹) → (((2nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵 ↔ (2nd ‘𝑟) ∈ 𝐵))
34	33	rabbiia 3185	. . . 4 ⊢ {𝑟 ∈ (𝐸 × 𝐹) ∣ ((2nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵} = {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd ‘𝑟) ∈ 𝐵}
35	31, 34	eqtri 2644	. . 3 ⊢ (◡(2nd ↾ (𝐸 × 𝐹)) “ 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd ‘𝑟) ∈ 𝐵}
36	27, 35	ineq12i 3812	. 2 ⊢ ((◡(1st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ (◡(2nd ↾ (𝐸 × 𝐹)) “ 𝐵)) = ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1st ‘𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd ‘𝑟) ∈ 𝐵})
37	19, 36	syl6eqr 2674	1 ⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → (𝐴 × 𝐵) = ((◡(1st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ (◡(2nd ↾ (𝐸 × 𝐹)) “ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574   × cxp 5112  ^◡ccnv 5113   ↾ cres 5116   “ cima 5117   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  1^st c1st 7166  2^nd c2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  cnre2csqima  29957  sxbrsigalem2  30348  sxbrsiga  30352