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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 4551 | . . 3 | |
2 | 1 | rneqi 4767 | . 2 |
3 | df-ima 4552 | . 2 | |
4 | df-ima 4552 | . . . . 5 | |
5 | df-res 4551 | . . . . . 6 | |
6 | 5 | rneqi 4767 | . . . . 5 |
7 | 4, 6 | eqtri 2160 | . . . 4 |
8 | 7 | ineq1i 3273 | . . 3 |
9 | cnvin 4946 | . . . . . 6 | |
10 | inxp 4673 | . . . . . . . . . 10 | |
11 | inv1 3399 | . . . . . . . . . . 11 | |
12 | incom 3268 | . . . . . . . . . . . 12 | |
13 | inv1 3399 | . . . . . . . . . . . 12 | |
14 | 12, 13 | eqtri 2160 | . . . . . . . . . . 11 |
15 | 11, 14 | xpeq12i 4561 | . . . . . . . . . 10 |
16 | 10, 15 | eqtr2i 2161 | . . . . . . . . 9 |
17 | 16 | ineq2i 3274 | . . . . . . . 8 |
18 | in32 3288 | . . . . . . . 8 | |
19 | xpindir 4675 | . . . . . . . . . . . 12 | |
20 | 19 | ineq2i 3274 | . . . . . . . . . . 11 |
21 | inass 3286 | . . . . . . . . . . 11 | |
22 | 20, 21 | eqtr4i 2163 | . . . . . . . . . 10 |
23 | 22 | ineq1i 3273 | . . . . . . . . 9 |
24 | inass 3286 | . . . . . . . . 9 | |
25 | 23, 24 | eqtri 2160 | . . . . . . . 8 |
26 | 17, 18, 25 | 3eqtr4i 2170 | . . . . . . 7 |
27 | 26 | cnveqi 4714 | . . . . . 6 |
28 | df-res 4551 | . . . . . . 7 | |
29 | cnvxp 4957 | . . . . . . . 8 | |
30 | 29 | ineq2i 3274 | . . . . . . 7 |
31 | 28, 30 | eqtr4i 2163 | . . . . . 6 |
32 | 9, 27, 31 | 3eqtr4ri 2171 | . . . . 5 |
33 | 32 | dmeqi 4740 | . . . 4 |
34 | incom 3268 | . . . . 5 | |
35 | dmres 4840 | . . . . 5 | |
36 | df-rn 4550 | . . . . . 6 | |
37 | 36 | ineq1i 3273 | . . . . 5 |
38 | 34, 35, 37 | 3eqtr4ri 2171 | . . . 4 |
39 | df-rn 4550 | . . . 4 | |
40 | 33, 38, 39 | 3eqtr4ri 2171 | . . 3 |
41 | 8, 40 | eqtr4i 2163 | . 2 |
42 | 2, 3, 41 | 3eqtr4i 2170 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1331 cvv 2686 cin 3070 cxp 4537 ccnv 4538 cdm 4539 crn 4540 cres 4541 cima 4542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 |
This theorem is referenced by: ecinxp 6504 |
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