Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > imasncls | Structured version Visualization version Unicode version |
Description: If a relation graph is closed, then an image set of a singleton is also closed. Corollary of proposition 4 of [BourbakiTop1] p. I.26. (Contributed by Thierry Arnoux, 14-Jan-2018.) |
Ref | Expression |
---|---|
imasnopn.1 | |
imasnopn.2 |
Ref | Expression |
---|---|
imasncls |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imasnopn.2 | . . . . . . 7 | |
2 | 1 | toptopon 20722 | . . . . . 6 TopOn |
3 | 2 | biimpi 206 | . . . . 5 TopOn |
4 | 3 | ad2antlr 763 | . . . 4 TopOn |
5 | imasnopn.1 | . . . . . . . 8 | |
6 | 5 | toptopon 20722 | . . . . . . 7 TopOn |
7 | 6 | biimpi 206 | . . . . . 6 TopOn |
8 | 7 | ad2antrr 762 | . . . . 5 TopOn |
9 | simprr 796 | . . . . 5 | |
10 | 4, 8, 9 | cnmptc 21465 | . . . 4 |
11 | 4 | cnmptid 21464 | . . . 4 |
12 | 4, 10, 11 | cnmpt1t 21468 | . . 3 |
13 | simprl 794 | . . . 4 | |
14 | 5, 1 | txuni 21395 | . . . . 5 |
15 | 14 | adantr 481 | . . . 4 |
16 | 13, 15 | sseqtrd 3641 | . . 3 |
17 | eqid 2622 | . . . 4 | |
18 | 17 | cncls2i 21074 | . . 3 |
19 | 12, 16, 18 | syl2anc 693 | . 2 |
20 | nfv 1843 | . . . . 5 | |
21 | nfcv 2764 | . . . . 5 | |
22 | nfrab1 3122 | . . . . 5 | |
23 | imass1 5500 | . . . . . . . . . . 11 | |
24 | 13, 23 | syl 17 | . . . . . . . . . 10 |
25 | xpimasn 5579 | . . . . . . . . . . 11 | |
26 | 25 | ad2antll 765 | . . . . . . . . . 10 |
27 | 24, 26 | sseqtrd 3641 | . . . . . . . . 9 |
28 | 27 | sseld 3602 | . . . . . . . 8 |
29 | 28 | pm4.71rd 667 | . . . . . . 7 |
30 | vex 3203 | . . . . . . . . . 10 | |
31 | elimasng 5491 | . . . . . . . . . 10 | |
32 | 30, 31 | mpan2 707 | . . . . . . . . 9 |
33 | 32 | ad2antll 765 | . . . . . . . 8 |
34 | 33 | anbi2d 740 | . . . . . . 7 |
35 | 29, 34 | bitrd 268 | . . . . . 6 |
36 | rabid 3116 | . . . . . 6 | |
37 | 35, 36 | syl6bbr 278 | . . . . 5 |
38 | 20, 21, 22, 37 | eqrd 3622 | . . . 4 |
39 | eqid 2622 | . . . . 5 | |
40 | 39 | mptpreima 5628 | . . . 4 |
41 | 38, 40 | syl6eqr 2674 | . . 3 |
42 | 41 | fveq2d 6195 | . 2 |
43 | nfcv 2764 | . . . 4 | |
44 | nfrab1 3122 | . . . 4 | |
45 | txtop 21372 | . . . . . . . . . . . . 13 | |
46 | 45 | adantr 481 | . . . . . . . . . . . 12 |
47 | 17 | clsss3 20863 | . . . . . . . . . . . 12 |
48 | 46, 16, 47 | syl2anc 693 | . . . . . . . . . . 11 |
49 | 48, 15 | sseqtr4d 3642 | . . . . . . . . . 10 |
50 | imass1 5500 | . . . . . . . . . 10 | |
51 | 49, 50 | syl 17 | . . . . . . . . 9 |
52 | 51, 26 | sseqtrd 3641 | . . . . . . . 8 |
53 | 52 | sseld 3602 | . . . . . . 7 |
54 | 53 | pm4.71rd 667 | . . . . . 6 |
55 | elimasng 5491 | . . . . . . . . 9 | |
56 | 30, 55 | mpan2 707 | . . . . . . . 8 |
57 | 56 | ad2antll 765 | . . . . . . 7 |
58 | 57 | anbi2d 740 | . . . . . 6 |
59 | 54, 58 | bitrd 268 | . . . . 5 |
60 | rabid 3116 | . . . . 5 | |
61 | 59, 60 | syl6bbr 278 | . . . 4 |
62 | 20, 43, 44, 61 | eqrd 3622 | . . 3 |
63 | 39 | mptpreima 5628 | . . 3 |
64 | 62, 63 | syl6eqr 2674 | . 2 |
65 | 19, 42, 64 | 3sstr4d 3648 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 crab 2916 cvv 3200 wss 3574 csn 4177 cop 4183 cuni 4436 cmpt 4729 cxp 5112 ccnv 5113 cima 5117 cfv 5888 (class class class)co 6650 ctop 20698 TopOnctopon 20715 ccl 20822 ccn 21028 ctx 21363 |
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-pow 4843 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-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 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-1st 7168 df-2nd 7169 df-map 7859 df-topgen 16104 df-top 20699 df-topon 20716 df-bases 20750 df-cld 20823 df-cls 20825 df-cn 21031 df-cnp 21032 df-tx 21365 |
This theorem is referenced by: utopreg 22056 |
Copyright terms: Public domain | W3C validator |