Theorem xpinpreima2 31143

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 5564	. . . . . 6 ⊢ (𝐸 × 𝐹) ⊆ (V × V)
2		rabss2 4052	. . . . . 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 769	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵))) → 𝑟 ∈ (V × V))
5		simpll 765	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵))) → 𝐴 ⊆ 𝐸)
6		simprrl 779	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵))) → (1st ‘𝑟) ∈ 𝐴)
7	5, 6	sseldd 3966	. . . . . . . 8 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵))) → (1st ‘𝑟) ∈ 𝐸)
8		simplr 767	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵))) → 𝐵 ⊆ 𝐹)
9		simprrr 780	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵))) → (2nd ‘𝑟) ∈ 𝐵)
10	8, 9	sseldd 3966	. . . . . . . 8 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵))) → (2nd ‘𝑟) ∈ 𝐹)
11	7, 10	jca 514	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵))) → ((1st ‘𝑟) ∈ 𝐸 ∧ (2nd ‘𝑟) ∈ 𝐹))
12		elxp7 7716	. . . . . . 7 ⊢ (𝑟 ∈ (𝐸 × 𝐹) ↔ (𝑟 ∈ (V × V) ∧ ((1st ‘𝑟) ∈ 𝐸 ∧ (2nd ‘𝑟) ∈ 𝐹)))
13	4, 11, 12	sylanbrc 585	. . . . . 6 ⊢ (((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) ∧ (𝑟 ∈ (V × V) ∧ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵))) → 𝑟 ∈ (𝐸 × 𝐹))
14	13	rabss3d 30268	. . . . 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 3982	. . . 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 7718	. . . 4 ⊢ (𝐴 × 𝐵) = {𝑟 ∈ (V × V) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)}
17	15, 16	syl6reqr 2873	. . 3 ⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → (𝐴 × 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st ‘𝑟) ∈ 𝐴 ∧ (2nd ‘𝑟) ∈ 𝐵)})
18		inrab 4273	. . 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 2872	. 2 ⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → (𝐴 × 𝐵) = ({𝑟 ∈ (𝐸 × 𝐹) ∣ (1st ‘𝑟) ∈ 𝐴} ∩ {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd ‘𝑟) ∈ 𝐵}))
20		f1stres 7705	. . . . 5 ⊢ (1st ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐸
21		ffn 6507	. . . . 5 ⊢ ((1st ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐸 → (1st ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹))
22		fncnvima2 6824	. . . . 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 6682	. . . . . 6 ⊢ (𝑟 ∈ (𝐸 × 𝐹) → ((1st ↾ (𝐸 × 𝐹))‘𝑟) = (1st ‘𝑟))
25	24	eleq1d 2895	. . . . 5 ⊢ (𝑟 ∈ (𝐸 × 𝐹) → (((1st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴 ↔ (1st ‘𝑟) ∈ 𝐴))
26	25	rabbiia 3471	. . . 4 ⊢ {𝑟 ∈ (𝐸 × 𝐹) ∣ ((1st ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐴} = {𝑟 ∈ (𝐸 × 𝐹) ∣ (1st ‘𝑟) ∈ 𝐴}
27	23, 26	eqtri 2842	. . 3 ⊢ (◡(1st ↾ (𝐸 × 𝐹)) “ 𝐴) = {𝑟 ∈ (𝐸 × 𝐹) ∣ (1st ‘𝑟) ∈ 𝐴}
28		f2ndres 7706	. . . . 5 ⊢ (2nd ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐹
29		ffn 6507	. . . . 5 ⊢ ((2nd ↾ (𝐸 × 𝐹)):(𝐸 × 𝐹)⟶𝐹 → (2nd ↾ (𝐸 × 𝐹)) Fn (𝐸 × 𝐹))
30		fncnvima2 6824	. . . . 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 6682	. . . . . 6 ⊢ (𝑟 ∈ (𝐸 × 𝐹) → ((2nd ↾ (𝐸 × 𝐹))‘𝑟) = (2nd ‘𝑟))
33	32	eleq1d 2895	. . . . 5 ⊢ (𝑟 ∈ (𝐸 × 𝐹) → (((2nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵 ↔ (2nd ‘𝑟) ∈ 𝐵))
34	33	rabbiia 3471	. . . 4 ⊢ {𝑟 ∈ (𝐸 × 𝐹) ∣ ((2nd ↾ (𝐸 × 𝐹))‘𝑟) ∈ 𝐵} = {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd ‘𝑟) ∈ 𝐵}
35	31, 34	eqtri 2842	. . 3 ⊢ (◡(2nd ↾ (𝐸 × 𝐹)) “ 𝐵) = {𝑟 ∈ (𝐸 × 𝐹) ∣ (2nd ‘𝑟) ∈ 𝐵}
36	27, 35	ineq12i 4185	. 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 2872	1 ⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → (𝐴 × 𝐵) = ((◡(1st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ (◡(2nd ↾ (𝐸 × 𝐹)) “ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1531   ∈ wcel 2108  {crab 3140  Vcvv 3493   ∩ cin 3933   ⊆ wss 3934   × cxp 5546  ^◡ccnv 5547   ↾ cres 5550   “ cima 5551   Fn wfn 6343  ⟶wf 6344  ‘cfv 6348  1^st c1st 7679  2^nd c2nd 7680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-1st 7681  df-2nd 7682

This theorem is referenced by:  cnre2csqima  31147  sxbrsigalem2  31537  sxbrsiga  31541