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Mirrors > Home > ILE Home > Th. List > imainrect | Unicode version |
Description: Image of a relation restricted to a rectangular region. (Contributed by Stefan O'Rear, 19-Feb-2015.) |
Ref | Expression |
---|---|
imainrect |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-res 4632 | . . 3 | |
2 | 1 | rneqi 4848 | . 2 |
3 | df-ima 4633 | . 2 | |
4 | df-ima 4633 | . . . . 5 | |
5 | df-res 4632 | . . . . . 6 | |
6 | 5 | rneqi 4848 | . . . . 5 |
7 | 4, 6 | eqtri 2196 | . . . 4 |
8 | 7 | ineq1i 3330 | . . 3 |
9 | cnvin 5028 | . . . . . 6 | |
10 | inxp 4754 | . . . . . . . . . 10 | |
11 | inv1 3457 | . . . . . . . . . . 11 | |
12 | incom 3325 | . . . . . . . . . . . 12 | |
13 | inv1 3457 | . . . . . . . . . . . 12 | |
14 | 12, 13 | eqtri 2196 | . . . . . . . . . . 11 |
15 | 11, 14 | xpeq12i 4642 | . . . . . . . . . 10 |
16 | 10, 15 | eqtr2i 2197 | . . . . . . . . 9 |
17 | 16 | ineq2i 3331 | . . . . . . . 8 |
18 | in32 3345 | . . . . . . . 8 | |
19 | xpindir 4756 | . . . . . . . . . . . 12 | |
20 | 19 | ineq2i 3331 | . . . . . . . . . . 11 |
21 | inass 3343 | . . . . . . . . . . 11 | |
22 | 20, 21 | eqtr4i 2199 | . . . . . . . . . 10 |
23 | 22 | ineq1i 3330 | . . . . . . . . 9 |
24 | inass 3343 | . . . . . . . . 9 | |
25 | 23, 24 | eqtri 2196 | . . . . . . . 8 |
26 | 17, 18, 25 | 3eqtr4i 2206 | . . . . . . 7 |
27 | 26 | cnveqi 4795 | . . . . . 6 |
28 | df-res 4632 | . . . . . . 7 | |
29 | cnvxp 5039 | . . . . . . . 8 | |
30 | 29 | ineq2i 3331 | . . . . . . 7 |
31 | 28, 30 | eqtr4i 2199 | . . . . . 6 |
32 | 9, 27, 31 | 3eqtr4ri 2207 | . . . . 5 |
33 | 32 | dmeqi 4821 | . . . 4 |
34 | incom 3325 | . . . . 5 | |
35 | dmres 4921 | . . . . 5 | |
36 | df-rn 4631 | . . . . . 6 | |
37 | 36 | ineq1i 3330 | . . . . 5 |
38 | 34, 35, 37 | 3eqtr4ri 2207 | . . . 4 |
39 | df-rn 4631 | . . . 4 | |
40 | 33, 38, 39 | 3eqtr4ri 2207 | . . 3 |
41 | 8, 40 | eqtr4i 2199 | . 2 |
42 | 2, 3, 41 | 3eqtr4i 2206 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1353 cvv 2735 cin 3126 cxp 4618 ccnv 4619 cdm 4620 crn 4621 cres 4622 cima 4623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-v 2737 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-br 3999 df-opab 4060 df-xp 4626 df-rel 4627 df-cnv 4628 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 |
This theorem is referenced by: ecinxp 6600 |
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