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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 “ (𝑌A)) ∩ B)

Proof of Theorem imainrect
StepHypRef Expression
1 df-res 4300 . . 3 ((𝐺 ∩ (A × B)) ↾ 𝑌) = ((𝐺 ∩ (A × B)) ∩ (𝑌 × V))
21rneqi 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 “ (𝑌A)) = ran (𝐺 ↾ (𝑌A))
5 df-res 4300 . . . . . 6 (𝐺 ↾ (𝑌A)) = (𝐺 ∩ ((𝑌A) × V))
65rneqi 4505 . . . . 5 ran (𝐺 ↾ (𝑌A)) = ran (𝐺 ∩ ((𝑌A) × V))
74, 6eqtri 2057 . . . 4 (𝐺 “ (𝑌A)) = ran (𝐺 ∩ ((𝑌A) × V))
87ineq1i 3128 . . 3 ((𝐺 “ (𝑌A)) ∩ B) = (ran (𝐺 ∩ ((𝑌A) × V)) ∩ B)
9 cnvin 4674 . . . . . 6 ((𝐺 ∩ ((𝑌A) × V)) ∩ (V × B)) = ((𝐺 ∩ ((𝑌A) × 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
1412, 13eqtri 2057 . . . . . . . . . . 11 (V ∩ B) = B
1511, 14xpeq12i 4310 . . . . . . . . . 10 ((A ∩ V) × (V ∩ B)) = (A × B)
1610, 15eqtr2i 2058 . . . . . . . . 9 (A × B) = ((A × V) ∩ (V × B))
1716ineq2i 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 ((𝑌A) × V) = ((𝑌 × V) ∩ (A × V))
2019ineq2i 3129 . . . . . . . . . . 11 (𝐺 ∩ ((𝑌A) × V)) = (𝐺 ∩ ((𝑌 × V) ∩ (A × V)))
21 inass 3141 . . . . . . . . . . 11 ((𝐺 ∩ (𝑌 × V)) ∩ (A × V)) = (𝐺 ∩ ((𝑌 × V) ∩ (A × V)))
2220, 21eqtr4i 2060 . . . . . . . . . 10 (𝐺 ∩ ((𝑌A) × V)) = ((𝐺 ∩ (𝑌 × V)) ∩ (A × V))
2322ineq1i 3128 . . . . . . . . 9 ((𝐺 ∩ ((𝑌A) × 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)))
2523, 24eqtri 2057 . . . . . . . 8 ((𝐺 ∩ ((𝑌A) × V)) ∩ (V × B)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((A × V) ∩ (V × B)))
2617, 18, 253eqtr4i 2067 . . . . . . 7 ((𝐺 ∩ (A × B)) ∩ (𝑌 × V)) = ((𝐺 ∩ ((𝑌A) × V)) ∩ (V × B))
2726cnveqi 4453 . . . . . 6 ((𝐺 ∩ (A × B)) ∩ (𝑌 × V)) = ((𝐺 ∩ ((𝑌A) × V)) ∩ (V × B))
28 df-res 4300 . . . . . . 7 ((𝐺 ∩ ((𝑌A) × V)) ↾ B) = ((𝐺 ∩ ((𝑌A) × V)) ∩ (B × V))
29 cnvxp 4685 . . . . . . . 8 (V × B) = (B × V)
3029ineq2i 3129 . . . . . . 7 ((𝐺 ∩ ((𝑌A) × V)) ∩ (V × B)) = ((𝐺 ∩ ((𝑌A) × V)) ∩ (B × V))
3128, 30eqtr4i 2060 . . . . . 6 ((𝐺 ∩ ((𝑌A) × V)) ↾ B) = ((𝐺 ∩ ((𝑌A) × V)) ∩ (V × B))
329, 27, 313eqtr4ri 2068 . . . . 5 ((𝐺 ∩ ((𝑌A) × V)) ↾ B) = ((𝐺 ∩ (A × B)) ∩ (𝑌 × V))
3332dmeqi 4479 . . . 4 dom ((𝐺 ∩ ((𝑌A) × V)) ↾ B) = dom ((𝐺 ∩ (A × B)) ∩ (𝑌 × V))
34 incom 3123 . . . . 5 (B ∩ dom (𝐺 ∩ ((𝑌A) × V))) = (dom (𝐺 ∩ ((𝑌A) × V)) ∩ B)
35 dmres 4575 . . . . 5 dom ((𝐺 ∩ ((𝑌A) × V)) ↾ B) = (B ∩ dom (𝐺 ∩ ((𝑌A) × V)))
36 df-rn 4299 . . . . . 6 ran (𝐺 ∩ ((𝑌A) × V)) = dom (𝐺 ∩ ((𝑌A) × V))
3736ineq1i 3128 . . . . 5 (ran (𝐺 ∩ ((𝑌A) × V)) ∩ B) = (dom (𝐺 ∩ ((𝑌A) × V)) ∩ B)
3834, 35, 373eqtr4ri 2068 . . . 4 (ran (𝐺 ∩ ((𝑌A) × V)) ∩ B) = dom ((𝐺 ∩ ((𝑌A) × V)) ↾ B)
39 df-rn 4299 . . . 4 ran ((𝐺 ∩ (A × B)) ∩ (𝑌 × V)) = dom ((𝐺 ∩ (A × B)) ∩ (𝑌 × V))
4033, 38, 393eqtr4ri 2068 . . 3 ran ((𝐺 ∩ (A × B)) ∩ (𝑌 × V)) = (ran (𝐺 ∩ ((𝑌A) × V)) ∩ B)
418, 40eqtr4i 2060 . 2 ((𝐺 “ (𝑌A)) ∩ B) = ran ((𝐺 ∩ (A × B)) ∩ (𝑌 × V))
422, 3, 413eqtr4i 2067 1 ((𝐺 ∩ (A × B)) “ 𝑌) = ((𝐺 “ (𝑌A)) ∩ B)
Colors of variables: wff set class
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-bnd 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
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