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Mirrors > Home > ILE Home > Th. List > resiexg | Unicode version |
Description: The existence of a restricted identity function, proved without using the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.) |
Ref | Expression |
---|---|
resiexg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 4847 | . . 3 | |
2 | simpr 109 | . . . . 5 | |
3 | eleq1 2202 | . . . . . 6 | |
4 | 3 | biimpa 294 | . . . . 5 |
5 | 2, 4 | jca 304 | . . . 4 |
6 | vex 2689 | . . . . . 6 | |
7 | 6 | opelres 4824 | . . . . 5 |
8 | df-br 3930 | . . . . . . 7 | |
9 | 6 | ideq 4691 | . . . . . . 7 |
10 | 8, 9 | bitr3i 185 | . . . . . 6 |
11 | 10 | anbi1i 453 | . . . . 5 |
12 | 7, 11 | bitri 183 | . . . 4 |
13 | opelxp 4569 | . . . 4 | |
14 | 5, 12, 13 | 3imtr4i 200 | . . 3 |
15 | 1, 14 | relssi 4630 | . 2 |
16 | xpexg 4653 | . . 3 | |
17 | 16 | anidms 394 | . 2 |
18 | ssexg 4067 | . 2 | |
19 | 15, 17, 18 | sylancr 410 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1480 cvv 2686 wss 3071 cop 3530 class class class wbr 3929 cid 4210 cxp 4537 cres 4541 |
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-13 1491 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 ax-un 4355 |
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-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-res 4551 |
This theorem is referenced by: ordiso 6921 omct 7002 ctssexmid 7024 ndxarg 11982 subctctexmid 13196 |
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