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Mirrors > Home > MPE Home > Th. List > residpr | Structured version Visualization version Unicode version |
Description: Restriction of the identity to a pair. (Contributed by AV, 11-Dec-2018.) |
Ref | Expression |
---|---|
residpr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4180 | . . . 4 | |
2 | 1 | reseq2i 5393 | . . 3 |
3 | resundi 5410 | . . 3 | |
4 | 2, 3 | eqtri 2644 | . 2 |
5 | xpsng 6406 | . . . . . 6 | |
6 | 5 | anidms 677 | . . . . 5 |
7 | 6 | adantr 481 | . . . 4 |
8 | xpsng 6406 | . . . . . 6 | |
9 | 8 | anidms 677 | . . . . 5 |
10 | 9 | adantl 482 | . . . 4 |
11 | 7, 10 | uneq12d 3768 | . . 3 |
12 | restidsing 5458 | . . . 4 | |
13 | restidsing 5458 | . . . 4 | |
14 | 12, 13 | uneq12i 3765 | . . 3 |
15 | df-pr 4180 | . . 3 | |
16 | 11, 14, 15 | 3eqtr4g 2681 | . 2 |
17 | 4, 16 | syl5eq 2668 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cun 3572 csn 4177 cpr 4179 cop 4183 cid 5023 cxp 5112 cres 5116 |
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-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 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-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 |
This theorem is referenced by: psgnprfval1 17942 |
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