Theorem imainrect 4709

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 4300	. . 3 ⊢ ((𝐺 ∩ (A × B)) ↾ 𝑌) = ((𝐺 ∩ (A × B)) ∩ (𝑌 × V))
2	1	rneqi 4505	. 2 ⊢ ran ((𝐺 ∩ (A × B)) ↾ 𝑌) = ran ((𝐺 ∩ (A × B)) ∩ (𝑌 × V))
3		df-ima 4301	. 2 ⊢ ((𝐺 ∩ (A × B)) “ 𝑌) = ran ((𝐺 ∩ (A × B)) ↾ 𝑌)
4		df-ima 4301	. . . . 5 ⊢ (𝐺 “ (𝑌 ∩ A)) = ran (𝐺 ↾ (𝑌 ∩ A))
5		df-res 4300	. . . . . 6 ⊢ (𝐺 ↾ (𝑌 ∩ A)) = (𝐺 ∩ ((𝑌 ∩ A) × V))
6	5	rneqi 4505	. . . . 5 ⊢ ran (𝐺 ↾ (𝑌 ∩ A)) = ran (𝐺 ∩ ((𝑌 ∩ A) × V))
7	4, 6	eqtri 2057	. . . 4 ⊢ (𝐺 “ (𝑌 ∩ A)) = ran (𝐺 ∩ ((𝑌 ∩ A) × V))
8	7	ineq1i 3128	. . 3 ⊢ ((𝐺 “ (𝑌 ∩ A)) ∩ B) = (ran (𝐺 ∩ ((𝑌 ∩ A) × V)) ∩ B)
9		cnvin 4674	. . . . . 6 ⊢ ◡((𝐺 ∩ ((𝑌 ∩ A) × V)) ∩ (V × B)) = (◡(𝐺 ∩ ((𝑌 ∩ A) × V)) ∩ ◡(V × B))
10		inxp 4413	. . . . . . . . . 10 ⊢ ((A × V) ∩ (V × B)) = ((A ∩ V) × (V ∩ B))
11		inv1 3247	. . . . . . . . . . 11 ⊢ (A ∩ V) = A
12		incom 3123	. . . . . . . . . . . 12 ⊢ (V ∩ B) = (B ∩ V)
13		inv1 3247	. . . . . . . . . . . 12 ⊢ (B ∩ V) = B
14	12, 13	eqtri 2057	. . . . . . . . . . 11 ⊢ (V ∩ B) = B
15	11, 14	xpeq12i 4310	. . . . . . . . . 10 ⊢ ((A ∩ V) × (V ∩ B)) = (A × B)
16	10, 15	eqtr2i 2058	. . . . . . . . 9 ⊢ (A × B) = ((A × V) ∩ (V × B))
17	16	ineq2i 3129	. . . . . . . 8 ⊢ ((𝐺 ∩ (𝑌 × V)) ∩ (A × B)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((A × V) ∩ (V × B)))
18		in32 3143	. . . . . . . 8 ⊢ ((𝐺 ∩ (A × B)) ∩ (𝑌 × V)) = ((𝐺 ∩ (𝑌 × V)) ∩ (A × B))
19		xpindir 4415	. . . . . . . . . . . 12 ⊢ ((𝑌 ∩ A) × V) = ((𝑌 × V) ∩ (A × V))
20	19	ineq2i 3129	. . . . . . . . . . 11 ⊢ (𝐺 ∩ ((𝑌 ∩ A) × V)) = (𝐺 ∩ ((𝑌 × V) ∩ (A × V)))
21		inass 3141	. . . . . . . . . . 11 ⊢ ((𝐺 ∩ (𝑌 × V)) ∩ (A × V)) = (𝐺 ∩ ((𝑌 × V) ∩ (A × V)))
22	20, 21	eqtr4i 2060	. . . . . . . . . 10 ⊢ (𝐺 ∩ ((𝑌 ∩ A) × V)) = ((𝐺 ∩ (𝑌 × V)) ∩ (A × V))
23	22	ineq1i 3128	. . . . . . . . 9 ⊢ ((𝐺 ∩ ((𝑌 ∩ A) × V)) ∩ (V × B)) = (((𝐺 ∩ (𝑌 × V)) ∩ (A × V)) ∩ (V × B))
24		inass 3141	. . . . . . . . 9 ⊢ (((𝐺 ∩ (𝑌 × V)) ∩ (A × V)) ∩ (V × B)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((A × V) ∩ (V × B)))
25	23, 24	eqtri 2057	. . . . . . . 8 ⊢ ((𝐺 ∩ ((𝑌 ∩ A) × V)) ∩ (V × B)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((A × V) ∩ (V × B)))
26	17, 18, 25	3eqtr4i 2067	. . . . . . 7 ⊢ ((𝐺 ∩ (A × B)) ∩ (𝑌 × V)) = ((𝐺 ∩ ((𝑌 ∩ A) × V)) ∩ (V × B))
27	26	cnveqi 4453	. . . . . 6 ⊢ ◡((𝐺 ∩ (A × B)) ∩ (𝑌 × V)) = ◡((𝐺 ∩ ((𝑌 ∩ A) × V)) ∩ (V × B))
28		df-res 4300	. . . . . . 7 ⊢ (◡(𝐺 ∩ ((𝑌 ∩ A) × V)) ↾ B) = (◡(𝐺 ∩ ((𝑌 ∩ A) × V)) ∩ (B × V))
29		cnvxp 4685	. . . . . . . 8 ⊢ ◡(V × B) = (B × V)
30	29	ineq2i 3129	. . . . . . 7 ⊢ (◡(𝐺 ∩ ((𝑌 ∩ A) × V)) ∩ ◡(V × B)) = (◡(𝐺 ∩ ((𝑌 ∩ A) × V)) ∩ (B × V))
31	28, 30	eqtr4i 2060	. . . . . 6 ⊢ (◡(𝐺 ∩ ((𝑌 ∩ A) × V)) ↾ B) = (◡(𝐺 ∩ ((𝑌 ∩ A) × V)) ∩ ◡(V × B))
32	9, 27, 31	3eqtr4ri 2068	. . . . 5 ⊢ (◡(𝐺 ∩ ((𝑌 ∩ A) × V)) ↾ B) = ◡((𝐺 ∩ (A × B)) ∩ (𝑌 × V))
33	32	dmeqi 4479	. . . 4 ⊢ dom (◡(𝐺 ∩ ((𝑌 ∩ A) × V)) ↾ B) = dom ◡((𝐺 ∩ (A × B)) ∩ (𝑌 × V))
34		incom 3123	. . . . 5 ⊢ (B ∩ dom ◡(𝐺 ∩ ((𝑌 ∩ A) × V))) = (dom ◡(𝐺 ∩ ((𝑌 ∩ A) × V)) ∩ B)
35		dmres 4575	. . . . 5 ⊢ dom (◡(𝐺 ∩ ((𝑌 ∩ A) × V)) ↾ B) = (B ∩ dom ◡(𝐺 ∩ ((𝑌 ∩ A) × V)))
36		df-rn 4299	. . . . . 6 ⊢ ran (𝐺 ∩ ((𝑌 ∩ A) × V)) = dom ◡(𝐺 ∩ ((𝑌 ∩ A) × V))
37	36	ineq1i 3128	. . . . 5 ⊢ (ran (𝐺 ∩ ((𝑌 ∩ A) × V)) ∩ B) = (dom ◡(𝐺 ∩ ((𝑌 ∩ A) × V)) ∩ B)
38	34, 35, 37	3eqtr4ri 2068	. . . 4 ⊢ (ran (𝐺 ∩ ((𝑌 ∩ A) × V)) ∩ B) = dom (◡(𝐺 ∩ ((𝑌 ∩ A) × V)) ↾ B)
39		df-rn 4299	. . . 4 ⊢ ran ((𝐺 ∩ (A × B)) ∩ (𝑌 × V)) = dom ◡((𝐺 ∩ (A × B)) ∩ (𝑌 × V))
40	33, 38, 39	3eqtr4ri 2068	. . 3 ⊢ ran ((𝐺 ∩ (A × B)) ∩ (𝑌 × V)) = (ran (𝐺 ∩ ((𝑌 ∩ A) × V)) ∩ B)
41	8, 40	eqtr4i 2060	. 2 ⊢ ((𝐺 “ (𝑌 ∩ A)) ∩ B) = ran ((𝐺 ∩ (A × B)) ∩ (𝑌 × V))
42	2, 3, 41	3eqtr4i 2067	1 ⊢ ((𝐺 ∩ (A × B)) “ 𝑌) = ((𝐺 “ (𝑌 ∩ A)) ∩ B)

Syntax hints:   = wceq 1242  Vcvv 2551   ∩ cin 2910   × cxp 4286  ^◡ccnv 4287  dom cdm 4288  ran crn 4289   ↾ cres 4290   “ cima 4291

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301

This theorem is referenced by:  ecinxp  6117