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Mirrors > Home > MPE Home > Th. List > restco | Structured version Visualization version Unicode version |
Description: Composition of subspaces. (Contributed by Mario Carneiro, 15-Dec-2013.) (Revised by Mario Carneiro, 1-May-2015.) |
Ref | Expression |
---|---|
restco | ↾t ↾t ↾t |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3203 | . . . . 5 | |
2 | 1 | inex1 4799 | . . . 4 |
3 | ineq1 3807 | . . . . 5 | |
4 | inass 3823 | . . . . 5 | |
5 | 3, 4 | syl6eq 2672 | . . . 4 |
6 | 2, 5 | abrexco 6502 | . . 3 |
7 | eqid 2622 | . . . . . 6 | |
8 | 7 | rnmpt 5371 | . . . . 5 |
9 | mpteq1 4737 | . . . . 5 | |
10 | 8, 9 | ax-mp 5 | . . . 4 |
11 | 10 | rnmpt 5371 | . . 3 |
12 | eqid 2622 | . . . 4 | |
13 | 12 | rnmpt 5371 | . . 3 |
14 | 6, 11, 13 | 3eqtr4i 2654 | . 2 |
15 | restval 16087 | . . . . 5 ↾t | |
16 | 15 | 3adant3 1081 | . . . 4 ↾t |
17 | 16 | oveq1d 6665 | . . 3 ↾t ↾t ↾t |
18 | ovex 6678 | . . . . 5 ↾t | |
19 | 16, 18 | syl6eqelr 2710 | . . . 4 |
20 | simp3 1063 | . . . 4 | |
21 | restval 16087 | . . . 4 ↾t | |
22 | 19, 20, 21 | syl2anc 693 | . . 3 ↾t |
23 | 17, 22 | eqtrd 2656 | . 2 ↾t ↾t |
24 | simp1 1061 | . . 3 | |
25 | inex1g 4801 | . . . 4 | |
26 | 25 | 3ad2ant2 1083 | . . 3 |
27 | restval 16087 | . . 3 ↾t | |
28 | 24, 26, 27 | syl2anc 693 | . 2 ↾t |
29 | 14, 23, 28 | 3eqtr4a 2682 | 1 ↾t ↾t ↾t |
Colors of variables: wff setvar class |
Syntax hints: wi 4 w3a 1037 wceq 1483 wcel 1990 cab 2608 wrex 2913 cvv 3200 cin 3573 cmpt 4729 crn 5115 (class class class)co 6650 ↾t crest 16081 |
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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
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-sbc 3436 df-csb 3534 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-uni 4437 df-iun 4522 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-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-rest 16083 |
This theorem is referenced by: restabs 20969 restin 20970 resstopn 20990 ressuss 22067 smfres 40997 |
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