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Theorem imainrect 4793
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 4384 . . 3 ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌) = ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
21rneqi 4589 . 2 ran ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
3 df-ima 4385 . 2 ((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ↾ 𝑌)
4 df-ima 4385 . . . . 5 (𝐺 “ (𝑌𝐴)) = ran (𝐺 ↾ (𝑌𝐴))
5 df-res 4384 . . . . . 6 (𝐺 ↾ (𝑌𝐴)) = (𝐺 ∩ ((𝑌𝐴) × V))
65rneqi 4589 . . . . 5 ran (𝐺 ↾ (𝑌𝐴)) = ran (𝐺 ∩ ((𝑌𝐴) × V))
74, 6eqtri 2076 . . . 4 (𝐺 “ (𝑌𝐴)) = ran (𝐺 ∩ ((𝑌𝐴) × V))
87ineq1i 3161 . . 3 ((𝐺 “ (𝑌𝐴)) ∩ 𝐵) = (ran (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵)
9 cnvin 4758 . . . . . 6 ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵))
10 inxp 4497 . . . . . . . . . 10 ((𝐴 × V) ∩ (V × 𝐵)) = ((𝐴 ∩ V) × (V ∩ 𝐵))
11 inv1 3280 . . . . . . . . . . 11 (𝐴 ∩ V) = 𝐴
12 incom 3156 . . . . . . . . . . . 12 (V ∩ 𝐵) = (𝐵 ∩ V)
13 inv1 3280 . . . . . . . . . . . 12 (𝐵 ∩ V) = 𝐵
1412, 13eqtri 2076 . . . . . . . . . . 11 (V ∩ 𝐵) = 𝐵
1511, 14xpeq12i 4394 . . . . . . . . . 10 ((𝐴 ∩ V) × (V ∩ 𝐵)) = (𝐴 × 𝐵)
1610, 15eqtr2i 2077 . . . . . . . . 9 (𝐴 × 𝐵) = ((𝐴 × V) ∩ (V × 𝐵))
1716ineq2i 3162 . . . . . . . 8 ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
18 in32 3176 . . . . . . . 8 ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × 𝐵))
19 xpindir 4499 . . . . . . . . . . . 12 ((𝑌𝐴) × V) = ((𝑌 × V) ∩ (𝐴 × V))
2019ineq2i 3162 . . . . . . . . . . 11 (𝐺 ∩ ((𝑌𝐴) × V)) = (𝐺 ∩ ((𝑌 × V) ∩ (𝐴 × V)))
21 inass 3174 . . . . . . . . . . 11 ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) = (𝐺 ∩ ((𝑌 × V) ∩ (𝐴 × V)))
2220, 21eqtr4i 2079 . . . . . . . . . 10 (𝐺 ∩ ((𝑌𝐴) × V)) = ((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V))
2322ineq1i 3161 . . . . . . . . 9 ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵)) = (((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) ∩ (V × 𝐵))
24 inass 3174 . . . . . . . . 9 (((𝐺 ∩ (𝑌 × V)) ∩ (𝐴 × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
2523, 24eqtri 2076 . . . . . . . 8 ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ (𝑌 × V)) ∩ ((𝐴 × V) ∩ (V × 𝐵)))
2617, 18, 253eqtr4i 2086 . . . . . . 7 ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵))
2726cnveqi 4537 . . . . . 6 ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵))
28 df-res 4384 . . . . . . 7 ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (𝐵 × V))
29 cnvxp 4769 . . . . . . . 8 (V × 𝐵) = (𝐵 × V)
3029ineq2i 3162 . . . . . . 7 ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵)) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (𝐵 × V))
3128, 30eqtr4i 2079 . . . . . 6 ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = ((𝐺 ∩ ((𝑌𝐴) × V)) ∩ (V × 𝐵))
329, 27, 313eqtr4ri 2087 . . . . 5 ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
3332dmeqi 4563 . . . 4 dom ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = dom ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
34 incom 3156 . . . . 5 (𝐵 ∩ dom (𝐺 ∩ ((𝑌𝐴) × V))) = (dom (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵)
35 dmres 4659 . . . . 5 dom ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵) = (𝐵 ∩ dom (𝐺 ∩ ((𝑌𝐴) × V)))
36 df-rn 4383 . . . . . 6 ran (𝐺 ∩ ((𝑌𝐴) × V)) = dom (𝐺 ∩ ((𝑌𝐴) × V))
3736ineq1i 3161 . . . . 5 (ran (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵) = (dom (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵)
3834, 35, 373eqtr4ri 2087 . . . 4 (ran (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵) = dom ((𝐺 ∩ ((𝑌𝐴) × V)) ↾ 𝐵)
39 df-rn 4383 . . . 4 ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = dom ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
4033, 38, 393eqtr4ri 2087 . . 3 ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V)) = (ran (𝐺 ∩ ((𝑌𝐴) × V)) ∩ 𝐵)
418, 40eqtr4i 2079 . 2 ((𝐺 “ (𝑌𝐴)) ∩ 𝐵) = ran ((𝐺 ∩ (𝐴 × 𝐵)) ∩ (𝑌 × V))
422, 3, 413eqtr4i 2086 1 ((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ((𝐺 “ (𝑌𝐴)) ∩ 𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1259  Vcvv 2574  cin 2943   × cxp 4370  ccnv 4371  dom cdm 4372  ran crn 4373  cres 4374  cima 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379  df-cnv 4380  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385
This theorem is referenced by:  ecinxp  6211
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