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Theorem imainrect 5685
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 5230 . . 3 ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌) = ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
21rneqi 5459 . 2 ran ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
3 df-ima 5231 . 2 ((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌)
4 df-ima 5231 . . . . 5 (𝐺 “ (𝑌𝐴)) = ran (𝐺 ↾ (𝑌𝐴))
5 df-res 5230 . . . . . 6 (𝐺 ↾ (𝑌𝐴)) = (𝐺 ∩ ((𝑌𝐴) × V))
65rneqi 5459 . . . . 5 ran (𝐺 ↾ (𝑌𝐴)) = ran (𝐺 ∩ ((𝑌𝐴) × V))
74, 6eqtri 2746 . . . 4 (𝐺 “ (𝑌𝐴)) = ran (𝐺 ∩ ((𝑌𝐴) × V))
87ineq1i 3918 . . 3 ((𝐺 “ (𝑌𝐴)) ∩ 𝐵) = (ran (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵)
9 cnvin 5650 . . . . . 6 ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵))
10 inxp 5362 . . . . . . . . . 10 ((𝐴 × V) ∩ (V × 𝐵)) = ((𝐴 ∩ V) × (V ∩ 𝐵))
11 inv1 4078 . . . . . . . . . . 11 (𝐴 ∩ V) = 𝐴
12 incom 3913 . . . . . . . . . . . 12 (V ∩ 𝐵) = (𝐵 ∩ V)
13 inv1 4078 . . . . . . . . . . . 12 (𝐵 ∩ V) = 𝐵
1412, 13eqtri 2746 . . . . . . . . . . 11 (V ∩ 𝐵) = 𝐵
1511, 14xpeq12i 5246 . . . . . . . . . 10 ((𝐴 ∩ V) × (V ∩ 𝐵)) = (𝐴 × 𝐵)
1610, 15eqtr2i 2747 . . . . . . . . 9 (𝐴 × 𝐵) = ((𝐴 × V) ∩ (V × 𝐵))
1716ineq2i 3919 . . . . . . . 8 ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
18 in32 3933 . . . . . . . 8 ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × 𝐵))
19 xpindir 5364 . . . . . . . . . . . 12 ((𝑌𝐴) × V) = ((𝑌 × V) ∩ (𝐴 × V))
2019ineq2i 3919 . . . . . . . . . . 11 (𝐺 ∩ ((𝑌𝐴) × V)) = (𝐺 ∩ ((𝑌 × V) ∩ (𝐴 × V)))
21 inass 3931 . . . . . . . . . . 11 ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) = (𝐺 ∩ ((𝑌 × V) ∩ (𝐴 × V)))
2220, 21eqtr4i 2749 . . . . . . . . . 10 (𝐺 ∩ ((𝑌𝐴) × V)) = ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V))
2322ineq1i 3918 . . . . . . . . 9 ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵)) = (((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) ∩ (V × 𝐵))
24 inass 3931 . . . . . . . . 9 (((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
2523, 24eqtri 2746 . . . . . . . 8 ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
2617, 18, 253eqtr4i 2756 . . . . . . 7 ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵))
2726cnveqi 5404 . . . . . 6 ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵))
28 df-res 5230 . . . . . . 7 ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (𝐵 × V))
29 cnvxp 5661 . . . . . . . 8 (V × 𝐵) = (𝐵 × V)
3029ineq2i 3919 . . . . . . 7 ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (𝐵 × V))
3128, 30eqtr4i 2749 . . . . . 6 ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵))
329, 27, 313eqtr4ri 2757 . . . . 5 ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
3332dmeqi 5432 . . . 4 dom ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = dom ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
34 incom 3913 . . . . 5 (𝐵 ∩ dom (𝐺 ∩ ((𝑌𝐴) × V))) = (dom (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵)
35 dmres 5529 . . . . 5 dom ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = (𝐵 ∩ dom (𝐺 ∩ ((𝑌𝐴) × V)))
36 df-rn 5229 . . . . . 6 ran (𝐺 ∩ ((𝑌𝐴) × V)) = dom (𝐺 ∩ ((𝑌𝐴) × V))
3736ineq1i 3918 . . . . 5 (ran (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵) = (dom (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵)
3834, 35, 373eqtr4ri 2757 . . . 4 (ran (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵) = dom ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵)
39 df-rn 5229 . . . 4 ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = dom ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
4033, 38, 393eqtr4ri 2757 . . 3 ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = (ran (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵)
418, 40eqtr4i 2749 . 2 ((𝐺 “ (𝑌𝐴)) ∩ 𝐵) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
422, 3, 413eqtr4i 2756 1 ((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ((𝐺 “ (𝑌𝐴)) ∩ 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1596  Vcvv 3304  cin 3679   × cxp 5216  ccnv 5217  dom cdm 5218  ran crn 5219  cres 5220  cima 5221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-xp 5224  df-rel 5225  df-cnv 5226  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231
This theorem is referenced by:  ecinxp  7940  marypha1lem  8455
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