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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 4610 | . . 3 | |
2 | 1 | rneqi 4826 | . 2 |
3 | df-ima 4611 | . 2 | |
4 | df-ima 4611 | . . . . 5 | |
5 | df-res 4610 | . . . . . 6 | |
6 | 5 | rneqi 4826 | . . . . 5 |
7 | 4, 6 | eqtri 2185 | . . . 4 |
8 | 7 | ineq1i 3314 | . . 3 |
9 | cnvin 5005 | . . . . . 6 | |
10 | inxp 4732 | . . . . . . . . . 10 | |
11 | inv1 3440 | . . . . . . . . . . 11 | |
12 | incom 3309 | . . . . . . . . . . . 12 | |
13 | inv1 3440 | . . . . . . . . . . . 12 | |
14 | 12, 13 | eqtri 2185 | . . . . . . . . . . 11 |
15 | 11, 14 | xpeq12i 4620 | . . . . . . . . . 10 |
16 | 10, 15 | eqtr2i 2186 | . . . . . . . . 9 |
17 | 16 | ineq2i 3315 | . . . . . . . 8 |
18 | in32 3329 | . . . . . . . 8 | |
19 | xpindir 4734 | . . . . . . . . . . . 12 | |
20 | 19 | ineq2i 3315 | . . . . . . . . . . 11 |
21 | inass 3327 | . . . . . . . . . . 11 | |
22 | 20, 21 | eqtr4i 2188 | . . . . . . . . . 10 |
23 | 22 | ineq1i 3314 | . . . . . . . . 9 |
24 | inass 3327 | . . . . . . . . 9 | |
25 | 23, 24 | eqtri 2185 | . . . . . . . 8 |
26 | 17, 18, 25 | 3eqtr4i 2195 | . . . . . . 7 |
27 | 26 | cnveqi 4773 | . . . . . 6 |
28 | df-res 4610 | . . . . . . 7 | |
29 | cnvxp 5016 | . . . . . . . 8 | |
30 | 29 | ineq2i 3315 | . . . . . . 7 |
31 | 28, 30 | eqtr4i 2188 | . . . . . 6 |
32 | 9, 27, 31 | 3eqtr4ri 2196 | . . . . 5 |
33 | 32 | dmeqi 4799 | . . . 4 |
34 | incom 3309 | . . . . 5 | |
35 | dmres 4899 | . . . . 5 | |
36 | df-rn 4609 | . . . . . 6 | |
37 | 36 | ineq1i 3314 | . . . . 5 |
38 | 34, 35, 37 | 3eqtr4ri 2196 | . . . 4 |
39 | df-rn 4609 | . . . 4 | |
40 | 33, 38, 39 | 3eqtr4ri 2196 | . . 3 |
41 | 8, 40 | eqtr4i 2188 | . 2 |
42 | 2, 3, 41 | 3eqtr4i 2195 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1342 cvv 2721 cin 3110 cxp 4596 ccnv 4597 cdm 4598 crn 4599 cres 4600 cima 4601 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 df-cnv 4606 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 |
This theorem is referenced by: ecinxp 6567 |
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