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Mirrors > Home > MPE Home > Th. List > ressn | Structured version Visualization version Unicode version |
Description: Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Ref | Expression |
---|---|
ressn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 5426 | . 2 | |
2 | relxp 5227 | . 2 | |
3 | ancom 466 | . . . 4 | |
4 | vex 3203 | . . . . . . 7 | |
5 | vex 3203 | . . . . . . 7 | |
6 | 4, 5 | elimasn 5490 | . . . . . 6 |
7 | elsni 4194 | . . . . . . . . 9 | |
8 | 7 | sneqd 4189 | . . . . . . . 8 |
9 | 8 | imaeq2d 5466 | . . . . . . 7 |
10 | 9 | eleq2d 2687 | . . . . . 6 |
11 | 6, 10 | syl5bbr 274 | . . . . 5 |
12 | 11 | pm5.32i 669 | . . . 4 |
13 | 3, 12 | bitri 264 | . . 3 |
14 | 5 | opelres 5401 | . . 3 |
15 | opelxp 5146 | . . 3 | |
16 | 13, 14, 15 | 3bitr4i 292 | . 2 |
17 | 1, 2, 16 | eqrelriiv 5214 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wa 384 wceq 1483 wcel 1990 csn 4177 cop 4183 cxp 5112 cres 5116 cima 5117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 |
This theorem is referenced by: gsum2dlem2 18370 dprd2da 18441 ustneism 22027 |
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