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Mirrors > Home > ILE Home > Th. List > cores2 | Unicode version |
Description: Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.) |
Ref | Expression |
---|---|
cores2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdm4 4726 | . . . . . 6 | |
2 | 1 | sseq1i 3118 | . . . . 5 |
3 | cores 5037 | . . . . 5 | |
4 | 2, 3 | sylbi 120 | . . . 4 |
5 | cnvco 4719 | . . . . 5 | |
6 | cocnvcnv1 5044 | . . . . 5 | |
7 | 5, 6 | eqtri 2158 | . . . 4 |
8 | cnvco 4719 | . . . 4 | |
9 | 4, 7, 8 | 3eqtr4g 2195 | . . 3 |
10 | 9 | cnveqd 4710 | . 2 |
11 | relco 5032 | . . 3 | |
12 | dfrel2 4984 | . . 3 | |
13 | 11, 12 | mpbi 144 | . 2 |
14 | relco 5032 | . . 3 | |
15 | dfrel2 4984 | . . 3 | |
16 | 14, 15 | mpbi 144 | . 2 |
17 | 10, 13, 16 | 3eqtr3g 2193 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 wss 3066 ccnv 4533 cdm 4534 crn 4535 cres 4536 ccom 4538 wrel 4539 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 |
This theorem is referenced by: cocnvres 5058 fcoi1 5298 |
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