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Theorem imainrect 6031
Description: Image by a restricted and corestricted binary relation (intersection of a binary relation with a Cartesian product). (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 5560 . . 3 ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌) = ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
21rneqi 5800 . 2 ran ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
3 df-ima 5561 . 2 ((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌)
4 df-ima 5561 . . . . 5 (𝐺 “ (𝑌𝐴)) = ran (𝐺 ↾ (𝑌𝐴))
5 df-res 5560 . . . . . 6 (𝐺 ↾ (𝑌𝐴)) = (𝐺 ∩ ((𝑌𝐴) × V))
65rneqi 5800 . . . . 5 ran (𝐺 ↾ (𝑌𝐴)) = ran (𝐺 ∩ ((𝑌𝐴) × V))
74, 6eqtri 2841 . . . 4 (𝐺 “ (𝑌𝐴)) = ran (𝐺 ∩ ((𝑌𝐴) × V))
87ineq1i 4182 . . 3 ((𝐺 “ (𝑌𝐴)) ∩ 𝐵) = (ran (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵)
9 cnvin 5996 . . . . . 6 ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵))
10 inxp 5696 . . . . . . . . . 10 ((𝐴 × V) ∩ (V × 𝐵)) = ((𝐴 ∩ V) × (V ∩ 𝐵))
11 inv1 4345 . . . . . . . . . . 11 (𝐴 ∩ V) = 𝐴
12 incom 4175 . . . . . . . . . . . 12 (V ∩ 𝐵) = (𝐵 ∩ V)
13 inv1 4345 . . . . . . . . . . . 12 (𝐵 ∩ V) = 𝐵
1412, 13eqtri 2841 . . . . . . . . . . 11 (V ∩ 𝐵) = 𝐵
1511, 14xpeq12i 5576 . . . . . . . . . 10 ((𝐴 ∩ V) × (V ∩ 𝐵)) = (𝐴 × 𝐵)
1610, 15eqtr2i 2842 . . . . . . . . 9 (𝐴 × 𝐵) = ((𝐴 × V) ∩ (V × 𝐵))
1716ineq2i 4183 . . . . . . . 8 ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
18 in32 4195 . . . . . . . 8 ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × 𝐵))
19 xpindir 5698 . . . . . . . . . . . 12 ((𝑌𝐴) × V) = ((𝑌 × V) ∩ (𝐴 × V))
2019ineq2i 4183 . . . . . . . . . . 11 (𝐺 ∩ ((𝑌𝐴) × V)) = (𝐺 ∩ ((𝑌 × V) ∩ (𝐴 × V)))
21 inass 4193 . . . . . . . . . . 11 ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) = (𝐺 ∩ ((𝑌 × V) ∩ (𝐴 × V)))
2220, 21eqtr4i 2844 . . . . . . . . . 10 (𝐺 ∩ ((𝑌𝐴) × V)) = ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V))
2322ineq1i 4182 . . . . . . . . 9 ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵)) = (((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) ∩ (V × 𝐵))
24 inass 4193 . . . . . . . . 9 (((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
2523, 24eqtri 2841 . . . . . . . 8 ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
2617, 18, 253eqtr4i 2851 . . . . . . 7 ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵))
2726cnveqi 5738 . . . . . 6 ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵))
28 df-res 5560 . . . . . . 7 ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (𝐵 × V))
29 cnvxp 6007 . . . . . . . 8 (V × 𝐵) = (𝐵 × V)
3029ineq2i 4183 . . . . . . 7 ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (𝐵 × V))
3128, 30eqtr4i 2844 . . . . . 6 ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵))
329, 27, 313eqtr4ri 2852 . . . . 5 ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
3332dmeqi 5766 . . . 4 dom ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = dom ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
34 incom 4175 . . . . 5 (𝐵 ∩ dom (𝐺 ∩ ((𝑌𝐴) × V))) = (dom (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵)
35 dmres 5868 . . . . 5 dom ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = (𝐵 ∩ dom (𝐺 ∩ ((𝑌𝐴) × V)))
36 df-rn 5559 . . . . . 6 ran (𝐺 ∩ ((𝑌𝐴) × V)) = dom (𝐺 ∩ ((𝑌𝐴) × V))
3736ineq1i 4182 . . . . 5 (ran (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵) = (dom (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵)
3834, 35, 373eqtr4ri 2852 . . . 4 (ran (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵) = dom ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵)
39 df-rn 5559 . . . 4 ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = dom ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
4033, 38, 393eqtr4ri 2852 . . 3 ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = (ran (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵)
418, 40eqtr4i 2844 . 2 ((𝐺 “ (𝑌𝐴)) ∩ 𝐵) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
422, 3, 413eqtr4i 2851 1 ((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ((𝐺 “ (𝑌𝐴)) ∩ 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  Vcvv 3492  cin 3932   × cxp 5546  ccnv 5547  dom cdm 5548  ran crn 5549  cres 5550  cima 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561
This theorem is referenced by:  ecinxp  8361  marypha1lem  8885
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