Theorem imainrect 5231

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 4784	. . 3 ⊢ ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌) = ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
2	1	rneqi 5008	. 2 ⊢ ran ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
3		df-ima 4785	. 2 ⊢ ((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌)
4		df-ima 4785	. . . . 5 ⊢ (𝐺 “ (𝑌 ∩ 𝐴)) = ran (𝐺 ↾ (𝑌 ∩ 𝐴))
5		df-res 4784	. . . . . 6 ⊢ (𝐺 ↾ (𝑌 ∩ 𝐴)) = (𝐺 ∩ ((𝑌 ∩ 𝐴) × V))
6	5	rneqi 5008	. . . . 5 ⊢ ran (𝐺 ↾ (𝑌 ∩ 𝐴)) = ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V))
7	4, 6	eqtri 2259	. . . 4 ⊢ (𝐺 “ (𝑌 ∩ 𝐴)) = ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V))
8	7	ineq1i 3428	. . 3 ⊢ ((𝐺 “ (𝑌 ∩ 𝐴)) ∩ 𝐵) = (ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵)
9		cnvin 5193	. . . . . 6 ⊢ ◡((𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (V × 𝐵)) = (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ ◡(V × 𝐵))
10		inxp 4912	. . . . . . . . . 10 ⊢ ((𝐴 × V) ∩ (V × 𝐵)) = ((𝐴 ∩ V) × (V ∩ 𝐵))
11		inv1 3559	. . . . . . . . . . 11 ⊢ (𝐴 ∩ V) = 𝐴
12		incom 3421	. . . . . . . . . . . 12 ⊢ (V ∩ 𝐵) = (𝐵 ∩ V)
13		inv1 3559	. . . . . . . . . . . 12 ⊢ (𝐵 ∩ V) = 𝐵
14	12, 13	eqtri 2259	. . . . . . . . . . 11 ⊢ (V ∩ 𝐵) = 𝐵
15	11, 14	xpeq12i 4794	. . . . . . . . . 10 ⊢ ((𝐴 ∩ V) × (V ∩ 𝐵)) = (𝐴 × 𝐵)
16	10, 15	eqtr2i 2260	. . . . . . . . 9 ⊢ (𝐴 × 𝐵) = ((𝐴 × V) ∩ (V × 𝐵))
17	16	ineq2i 3429	. . . . . . . 8 ⊢ ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
18		in32 3443	. . . . . . . 8 ⊢ ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × 𝐵))
19		xpindir 4914	. . . . . . . . . . . 12 ⊢ ((𝑌 ∩ 𝐴) × V) = ((𝑌 × V) ∩ (𝐴 × V))
20	19	ineq2i 3429	. . . . . . . . . . 11 ⊢ (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) = (𝐺 ∩ ((𝑌 × V) ∩ (𝐴 × V)))
21		inass 3441	. . . . . . . . . . 11 ⊢ ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) = (𝐺 ∩ ((𝑌 × V) ∩ (𝐴 × V)))
22	20, 21	eqtr4i 2262	. . . . . . . . . 10 ⊢ (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) = ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V))
23	22	ineq1i 3428	. . . . . . . . 9 ⊢ ((𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (V × 𝐵)) = (((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) ∩ (V × 𝐵))
24		inass 3441	. . . . . . . . 9 ⊢ (((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
25	23, 24	eqtri 2259	. . . . . . . 8 ⊢ ((𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
26	17, 18, 25	3eqtr4i 2269	. . . . . . 7 ⊢ ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (V × 𝐵))
27	26	cnveqi 4953	. . . . . 6 ⊢ ◡((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ◡((𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (V × 𝐵))
28		df-res 4784	. . . . . . 7 ⊢ (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵) = (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (𝐵 × V))
29		cnvxp 5204	. . . . . . . 8 ⊢ ◡(V × 𝐵) = (𝐵 × V)
30	29	ineq2i 3429	. . . . . . 7 ⊢ (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ ◡(V × 𝐵)) = (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ (𝐵 × V))
31	28, 30	eqtr4i 2262	. . . . . 6 ⊢ (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵) = (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ ◡(V × 𝐵))
32	9, 27, 31	3eqtr4ri 2270	. . . . 5 ⊢ (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵) = ◡((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
33	32	dmeqi 4980	. . . 4 ⊢ dom (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵) = dom ◡((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
34		incom 3421	. . . . 5 ⊢ (𝐵 ∩ dom ◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V))) = (dom ◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵)
35		dmres 5082	. . . . 5 ⊢ dom (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵) = (𝐵 ∩ dom ◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)))
36		df-rn 4783	. . . . . 6 ⊢ ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) = dom ◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V))
37	36	ineq1i 3428	. . . . 5 ⊢ (ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵) = (dom ◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵)
38	34, 35, 37	3eqtr4ri 2270	. . . 4 ⊢ (ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵) = dom (◡(𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ↾ 𝐵)
39		df-rn 4783	. . . 4 ⊢ ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = dom ◡((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
40	33, 38, 39	3eqtr4ri 2270	. . 3 ⊢ ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = (ran (𝐺 ∩ ((𝑌 ∩ 𝐴) × V)) ∩ 𝐵)
41	8, 40	eqtr4i 2262	. 2 ⊢ ((𝐺 “ (𝑌 ∩ 𝐴)) ∩ 𝐵) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
42	2, 3, 41	3eqtr4i 2269	1 ⊢ ((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ((𝐺 “ (𝑌 ∩ 𝐴)) ∩ 𝐵)

Syntax hints:   = wceq 1402  Vcvv 2821   ∩ cin 3219   × cxp 4770  ^◡ccnv 4771  dom cdm 4772  ran crn 4773   ↾ cres 4774   “ cima 4775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-xp 4778  df-rel 4779  df-cnv 4780  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785

This theorem is referenced by:  ecinxp  6878