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Theorem imainrect 6170
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 5678 . . 3 ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌) = ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
21rneqi 5926 . 2 ran ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
3 df-ima 5679 . 2 ((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌)
4 df-ima 5679 . . . . 5 (𝐺 “ (𝑌𝐴)) = ran (𝐺 ↾ (𝑌𝐴))
5 df-res 5678 . . . . . 6 (𝐺 ↾ (𝑌𝐴)) = (𝐺 ∩ ((𝑌𝐴) × V))
65rneqi 5926 . . . . 5 ran (𝐺 ↾ (𝑌𝐴)) = ran (𝐺 ∩ ((𝑌𝐴) × V))
74, 6eqtri 2752 . . . 4 (𝐺 “ (𝑌𝐴)) = ran (𝐺 ∩ ((𝑌𝐴) × V))
87ineq1i 4200 . . 3 ((𝐺 “ (𝑌𝐴)) ∩ 𝐵) = (ran (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵)
9 cnvin 6134 . . . . . 6 ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵))
10 inxp 5822 . . . . . . . . . 10 ((𝐴 × V) ∩ (V × 𝐵)) = ((𝐴 ∩ V) × (V ∩ 𝐵))
11 inv1 4386 . . . . . . . . . . 11 (𝐴 ∩ V) = 𝐴
12 incom 4193 . . . . . . . . . . . 12 (V ∩ 𝐵) = (𝐵 ∩ V)
13 inv1 4386 . . . . . . . . . . . 12 (𝐵 ∩ V) = 𝐵
1412, 13eqtri 2752 . . . . . . . . . . 11 (V ∩ 𝐵) = 𝐵
1511, 14xpeq12i 5694 . . . . . . . . . 10 ((𝐴 ∩ V) × (V ∩ 𝐵)) = (𝐴 × 𝐵)
1610, 15eqtr2i 2753 . . . . . . . . 9 (𝐴 × 𝐵) = ((𝐴 × V) ∩ (V × 𝐵))
1716ineq2i 4201 . . . . . . . 8 ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
18 in32 4213 . . . . . . . 8 ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × 𝐵))
19 xpindir 5824 . . . . . . . . . . . 12 ((𝑌𝐴) × V) = ((𝑌 × V) ∩ (𝐴 × V))
2019ineq2i 4201 . . . . . . . . . . 11 (𝐺 ∩ ((𝑌𝐴) × V)) = (𝐺 ∩ ((𝑌 × V) ∩ (𝐴 × V)))
21 inass 4211 . . . . . . . . . . 11 ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) = (𝐺 ∩ ((𝑌 × V) ∩ (𝐴 × V)))
2220, 21eqtr4i 2755 . . . . . . . . . 10 (𝐺 ∩ ((𝑌𝐴) × V)) = ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V))
2322ineq1i 4200 . . . . . . . . 9 ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵)) = (((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) ∩ (V × 𝐵))
24 inass 4211 . . . . . . . . 9 (((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
2523, 24eqtri 2752 . . . . . . . 8 ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
2617, 18, 253eqtr4i 2762 . . . . . . 7 ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵))
2726cnveqi 5864 . . . . . 6 ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵))
28 df-res 5678 . . . . . . 7 ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (𝐵 × V))
29 cnvxp 6146 . . . . . . . 8 (V × 𝐵) = (𝐵 × V)
3029ineq2i 4201 . . . . . . 7 ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (𝐵 × V))
3128, 30eqtr4i 2755 . . . . . 6 ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵))
329, 27, 313eqtr4ri 2763 . . . . 5 ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
3332dmeqi 5894 . . . 4 dom ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = dom ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
34 incom 4193 . . . . 5 (𝐵 ∩ dom (𝐺 ∩ ((𝑌𝐴) × V))) = (dom (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵)
35 dmres 5993 . . . . 5 dom ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = (𝐵 ∩ dom (𝐺 ∩ ((𝑌𝐴) × V)))
36 df-rn 5677 . . . . . 6 ran (𝐺 ∩ ((𝑌𝐴) × V)) = dom (𝐺 ∩ ((𝑌𝐴) × V))
3736ineq1i 4200 . . . . 5 (ran (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵) = (dom (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵)
3834, 35, 373eqtr4ri 2763 . . . 4 (ran (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵) = dom ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵)
39 df-rn 5677 . . . 4 ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = dom ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
4033, 38, 393eqtr4ri 2763 . . 3 ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = (ran (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵)
418, 40eqtr4i 2755 . 2 ((𝐺 “ (𝑌𝐴)) ∩ 𝐵) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
422, 3, 413eqtr4i 2762 1 ((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ((𝐺 “ (𝑌𝐴)) ∩ 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  Vcvv 3466  cin 3939   × cxp 5664  ccnv 5665  dom cdm 5666  ran crn 5667  cres 5668  cima 5669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679
This theorem is referenced by:  ecinxp  8782  marypha1lem  9424
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