Theorem imainrect 4984

Description: Image of a relation restricted to a rectangular region. (Contributed by Stefan O'Rear, 19-Feb-2015.)

Assertion

Ref	Expression
imainrect	⊢ ((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ((𝐺 “ (𝑌 ∩ 𝐴)) ∩ 𝐵)

Proof of Theorem imainrect

Step	Hyp	Ref	Expression
1		df-res 4551	. . 3 ⊢ ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌) = ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
2	1	rneqi 4767	. 2 ⊢ ran ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
3		df-ima 4552	. 2 ⊢ ((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌)
4		df-ima 4552	. . . . 5 ⊢ (𝐺 “ (𝑌 ∩ 𝐴)) = ran (𝐺 ↾ (𝑌 ∩ 𝐴))
5		df-res 4551	. . . . . 6 ⊢ (𝐺 ↾ (𝑌 ∩ 𝐴)) = (𝐺 ∩ ((𝑌 ∩ 𝐴) × V))
6	5	rneqi 4767	. . . . 5 ⊢ ran (𝐺 ↾ (𝑌 ∩ 𝐴)) = ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V))
7	4, 6	eqtri 2160	. . . 4 ⊢ (𝐺 “ (𝑌 ∩ 𝐴)) = ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V))
8	7	ineq1i 3273	. . 3 ⊢ ((𝐺 “ (𝑌 ∩ 𝐴)) ∩ 𝐵) = (ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵)
9		cnvin 4946	. . . . . 6 ⊢ ◡((𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (V × 𝐵)) = (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ ◡(V × 𝐵))
10		inxp 4673	. . . . . . . . . 10 ⊢ ((𝐴 × V) ∩ (V × 𝐵)) = ((𝐴 ∩ V) × (V ∩ 𝐵))
11		inv1 3399	. . . . . . . . . . 11 ⊢ (𝐴 ∩ V) = 𝐴
12		incom 3268	. . . . . . . . . . . 12 ⊢ (V ∩ 𝐵) = (𝐵 ∩ V)
13		inv1 3399	. . . . . . . . . . . 12 ⊢ (𝐵 ∩ V) = 𝐵
14	12, 13	eqtri 2160	. . . . . . . . . . 11 ⊢ (V ∩ 𝐵) = 𝐵
15	11, 14	xpeq12i 4561	. . . . . . . . . 10 ⊢ ((𝐴 ∩ V) × (V ∩ 𝐵)) = (𝐴 × 𝐵)
16	10, 15	eqtr2i 2161	. . . . . . . . 9 ⊢ (𝐴 × 𝐵) = ((𝐴 × V) ∩ (V × 𝐵))
17	16	ineq2i 3274	. . . . . . . 8 ⊢ ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
18		in32 3288	. . . . . . . 8 ⊢ ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × 𝐵))
19		xpindir 4675	. . . . . . . . . . . 12 ⊢ ((𝑌 ∩ 𝐴) × V) = ((𝑌 × V) ∩ (𝐴 × V))
20	19	ineq2i 3274	. . . . . . . . . . 11 ⊢ (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) = (𝐺 ∩ ((𝑌 × V) ∩ (𝐴 × V)))
21		inass 3286	. . . . . . . . . . 11 ⊢ ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) = (𝐺 ∩ ((𝑌 × V) ∩ (𝐴 × V)))
22	20, 21	eqtr4i 2163	. . . . . . . . . 10 ⊢ (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) = ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V))
23	22	ineq1i 3273	. . . . . . . . 9 ⊢ ((𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (V × 𝐵)) = (((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) ∩ (V × 𝐵))
24		inass 3286	. . . . . . . . 9 ⊢ (((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
25	23, 24	eqtri 2160	. . . . . . . 8 ⊢ ((𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
26	17, 18, 25	3eqtr4i 2170	. . . . . . 7 ⊢ ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (V × 𝐵))
27	26	cnveqi 4714	. . . . . 6 ⊢ ◡((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ◡((𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (V × 𝐵))
28		df-res 4551	. . . . . . 7 ⊢ (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵) = (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (𝐵 × V))
29		cnvxp 4957	. . . . . . . 8 ⊢ ◡(V × 𝐵) = (𝐵 × V)
30	29	ineq2i 3274	. . . . . . 7 ⊢ (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ ◡(V × 𝐵)) = (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (𝐵 × V))
31	28, 30	eqtr4i 2163	. . . . . 6 ⊢ (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵) = (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ ◡(V × 𝐵))
32	9, 27, 31	3eqtr4ri 2171	. . . . 5 ⊢ (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵) = ◡((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
33	32	dmeqi 4740	. . . 4 ⊢ dom (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵) = dom ◡((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
34		incom 3268	. . . . 5 ⊢ (𝐵 ∩ dom ◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V))) = (dom ◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵)
35		dmres 4840	. . . . 5 ⊢ dom (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵) = (𝐵 ∩ dom ◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)))
36		df-rn 4550	. . . . . 6 ⊢ ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) = dom ◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V))
37	36	ineq1i 3273	. . . . 5 ⊢ (ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵) = (dom ◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵)
38	34, 35, 37	3eqtr4ri 2171	. . . 4 ⊢ (ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵) = dom (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵)
39		df-rn 4550	. . . 4 ⊢ ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = dom ◡((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
40	33, 38, 39	3eqtr4ri 2171	. . 3 ⊢ ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = (ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵)
41	8, 40	eqtr4i 2163	. 2 ⊢ ((𝐺 “ (𝑌 ∩ 𝐴)) ∩ 𝐵) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
42	2, 3, 41	3eqtr4i 2170	1 ⊢ ((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ((𝐺 “ (𝑌 ∩ 𝐴)) ∩ 𝐵)

Syntax hints:   = wceq 1331  Vcvv 2686   ∩ cin 3070   × cxp 4537  ^◡ccnv 4538  dom cdm 4539  ran crn 4540   ↾ cres 4541   “ cima 4542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552

This theorem is referenced by:  ecinxp  6504