Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapssbi | Structured version Visualization version Unicode version |
Description: Subset inheritance for set exponentiation. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
mapssbi.a | |
mapssbi.b | |
mapssbi.c | |
mapssbi.n |
Ref | Expression |
---|---|
mapssbi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapssbi.b | . . . . 5 | |
2 | 1 | adantr 481 | . . . 4 |
3 | simpr 477 | . . . 4 | |
4 | mapss 7900 | . . . 4 | |
5 | 2, 3, 4 | syl2anc 693 | . . 3 |
6 | 5 | ex 450 | . 2 |
7 | simplr 792 | . . . 4 | |
8 | nssrex 39260 | . . . . . . . 8 | |
9 | 8 | biimpi 206 | . . . . . . 7 |
10 | 9 | adantl 482 | . . . . . 6 |
11 | fconst6g 6094 | . . . . . . . . . . . . 13 | |
12 | 11 | adantl 482 | . . . . . . . . . . . 12 |
13 | mapssbi.a | . . . . . . . . . . . . . 14 | |
14 | mapssbi.c | . . . . . . . . . . . . . 14 | |
15 | elmapg 7870 | . . . . . . . . . . . . . 14 | |
16 | 13, 14, 15 | syl2anc 693 | . . . . . . . . . . . . 13 |
17 | 16 | adantr 481 | . . . . . . . . . . . 12 |
18 | 12, 17 | mpbird 247 | . . . . . . . . . . 11 |
19 | 18 | 3adant3 1081 | . . . . . . . . . 10 |
20 | 14 | adantr 481 | . . . . . . . . . . . . . 14 |
21 | 1 | adantr 481 | . . . . . . . . . . . . . 14 |
22 | mapssbi.n | . . . . . . . . . . . . . . 15 | |
23 | 22 | adantr 481 | . . . . . . . . . . . . . 14 |
24 | simpr 477 | . . . . . . . . . . . . . 14 | |
25 | 20, 21, 23, 24 | snelmap 39254 | . . . . . . . . . . . . 13 |
26 | 25 | adantlr 751 | . . . . . . . . . . . 12 |
27 | simplr 792 | . . . . . . . . . . . 12 | |
28 | 26, 27 | pm2.65da 600 | . . . . . . . . . . 11 |
29 | 28 | 3adant2 1080 | . . . . . . . . . 10 |
30 | nelss 3664 | . . . . . . . . . 10 | |
31 | 19, 29, 30 | syl2anc 693 | . . . . . . . . 9 |
32 | 31 | 3exp 1264 | . . . . . . . 8 |
33 | 32 | adantr 481 | . . . . . . 7 |
34 | 33 | rexlimdv 3030 | . . . . . 6 |
35 | 10, 34 | mpd 15 | . . . . 5 |
36 | 35 | adantlr 751 | . . . 4 |
37 | 7, 36 | condan 835 | . . 3 |
38 | 37 | ex 450 | . 2 |
39 | 6, 38 | impbid 202 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 w3a 1037 wcel 1990 wne 2794 wrex 2913 wss 3574 c0 3915 csn 4177 cxp 5112 wf 5884 (class class class)co 6650 cmap 7857 |
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-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-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-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-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 |
This theorem is referenced by: (None) |
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