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| Mirrors > Home > ILE Home > Th. List > imai | Unicode version | ||
| Description: Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.) |
| Ref | Expression |
|---|---|
| imai |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfima3 5024 |
. 2
| |
| 2 | df-br 4044 |
. . . . . . . 8
| |
| 3 | vex 2774 |
. . . . . . . . 9
| |
| 4 | 3 | ideq 4829 |
. . . . . . . 8
|
| 5 | 2, 4 | bitr3i 186 |
. . . . . . 7
|
| 6 | 5 | anbi2i 457 |
. . . . . 6
|
| 7 | ancom 266 |
. . . . . 6
| |
| 8 | 6, 7 | bitri 184 |
. . . . 5
|
| 9 | 8 | exbii 1627 |
. . . 4
|
| 10 | eleq1 2267 |
. . . . 5
| |
| 11 | 3, 10 | ceqsexv 2810 |
. . . 4
|
| 12 | 9, 11 | bitri 184 |
. . 3
|
| 13 | 12 | abbii 2320 |
. 2
|
| 14 | abid2 2325 |
. 2
| |
| 15 | 1, 13, 14 | 3eqtri 2229 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| 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 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ral 2488 df-rex 2489 df-v 2773 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-br 4044 df-opab 4105 df-id 4339 df-xp 4680 df-rel 4681 df-cnv 4682 df-dm 4684 df-rn 4685 df-res 4686 df-ima 4687 |
| This theorem is referenced by: rnresi 5038 cnvresid 5347 ecidsn 6668 |
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