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Theorem imainrect 5044
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 4611 . . 3 ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌) = ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
21rneqi 4827 . 2 ran ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
3 df-ima 4612 . 2 ((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌)
4 df-ima 4612 . . . . 5 (𝐺 “ (𝑌𝐴)) = ran (𝐺 ↾ (𝑌𝐴))
5 df-res 4611 . . . . . 6 (𝐺 ↾ (𝑌𝐴)) = (𝐺 ∩ ((𝑌𝐴) × V))
65rneqi 4827 . . . . 5 ran (𝐺 ↾ (𝑌𝐴)) = ran (𝐺 ∩ ((𝑌𝐴) × V))
74, 6eqtri 2185 . . . 4 (𝐺 “ (𝑌𝐴)) = ran (𝐺 ∩ ((𝑌𝐴) × V))
87ineq1i 3315 . . 3 ((𝐺 “ (𝑌𝐴)) ∩ 𝐵) = (ran (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵)
9 cnvin 5006 . . . . . 6 ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵))
10 inxp 4733 . . . . . . . . . 10 ((𝐴 × V) ∩ (V × 𝐵)) = ((𝐴 ∩ V) × (V ∩ 𝐵))
11 inv1 3441 . . . . . . . . . . 11 (𝐴 ∩ V) = 𝐴
12 incom 3310 . . . . . . . . . . . 12 (V ∩ 𝐵) = (𝐵 ∩ V)
13 inv1 3441 . . . . . . . . . . . 12 (𝐵 ∩ V) = 𝐵
1412, 13eqtri 2185 . . . . . . . . . . 11 (V ∩ 𝐵) = 𝐵
1511, 14xpeq12i 4621 . . . . . . . . . 10 ((𝐴 ∩ V) × (V ∩ 𝐵)) = (𝐴 × 𝐵)
1610, 15eqtr2i 2186 . . . . . . . . 9 (𝐴 × 𝐵) = ((𝐴 × V) ∩ (V × 𝐵))
1716ineq2i 3316 . . . . . . . 8 ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
18 in32 3330 . . . . . . . 8 ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × 𝐵))
19 xpindir 4735 . . . . . . . . . . . 12 ((𝑌𝐴) × V) = ((𝑌 × V) ∩ (𝐴 × V))
2019ineq2i 3316 . . . . . . . . . . 11 (𝐺 ∩ ((𝑌𝐴) × V)) = (𝐺 ∩ ((𝑌 × V) ∩ (𝐴 × V)))
21 inass 3328 . . . . . . . . . . 11 ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) = (𝐺 ∩ ((𝑌 × V) ∩ (𝐴 × V)))
2220, 21eqtr4i 2188 . . . . . . . . . 10 (𝐺 ∩ ((𝑌𝐴) × V)) = ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V))
2322ineq1i 3315 . . . . . . . . 9 ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵)) = (((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) ∩ (V × 𝐵))
24 inass 3328 . . . . . . . . 9 (((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
2523, 24eqtri 2185 . . . . . . . 8 ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
2617, 18, 253eqtr4i 2195 . . . . . . 7 ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵))
2726cnveqi 4774 . . . . . 6 ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵))
28 df-res 4611 . . . . . . 7 ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (𝐵 × V))
29 cnvxp 5017 . . . . . . . 8 (V × 𝐵) = (𝐵 × V)
3029ineq2i 3316 . . . . . . 7 ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (𝐵 × V))
3128, 30eqtr4i 2188 . . . . . 6 ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵))
329, 27, 313eqtr4ri 2196 . . . . 5 ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
3332dmeqi 4800 . . . 4 dom ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = dom ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
34 incom 3310 . . . . 5 (𝐵 ∩ dom (𝐺 ∩ ((𝑌𝐴) × V))) = (dom (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵)
35 dmres 4900 . . . . 5 dom ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = (𝐵 ∩ dom (𝐺 ∩ ((𝑌𝐴) × V)))
36 df-rn 4610 . . . . . 6 ran (𝐺 ∩ ((𝑌𝐴) × V)) = dom (𝐺 ∩ ((𝑌𝐴) × V))
3736ineq1i 3315 . . . . 5 (ran (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵) = (dom (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵)
3834, 35, 373eqtr4ri 2196 . . . 4 (ran (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵) = dom ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵)
39 df-rn 4610 . . . 4 ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = dom ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
4033, 38, 393eqtr4ri 2196 . . 3 ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = (ran (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵)
418, 40eqtr4i 2188 . 2 ((𝐺 “ (𝑌𝐴)) ∩ 𝐵) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
422, 3, 413eqtr4i 2195 1 ((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ((𝐺 “ (𝑌𝐴)) ∩ 𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1342  Vcvv 2722  cin 3111   × cxp 4597  ccnv 4598  dom cdm 4599  ran crn 4600  cres 4601  cima 4602
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2724  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-br 3978  df-opab 4039  df-xp 4605  df-rel 4606  df-cnv 4607  df-dm 4609  df-rn 4610  df-res 4611  df-ima 4612
This theorem is referenced by:  ecinxp  6568
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