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Mirrors > Home > MPE Home > Th. List > isssc | Structured version Visualization version Unicode version |
Description: Value of the subcategory subset relation when the arguments are known functions. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
isssc.1 | |
isssc.2 | |
isssc.3 |
Ref | Expression |
---|---|
isssc | cat |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brssc 16474 | . . . 4 cat | |
2 | fndm 5990 | . . . . . . . . . . . 12 | |
3 | 2 | adantl 482 | . . . . . . . . . . 11 |
4 | isssc.2 | . . . . . . . . . . . . 13 | |
5 | 4 | adantr 481 | . . . . . . . . . . . 12 |
6 | fndm 5990 | . . . . . . . . . . . 12 | |
7 | 5, 6 | syl 17 | . . . . . . . . . . 11 |
8 | 3, 7 | eqtr3d 2658 | . . . . . . . . . 10 |
9 | 8 | dmeqd 5326 | . . . . . . . . 9 |
10 | dmxpid 5345 | . . . . . . . . 9 | |
11 | dmxpid 5345 | . . . . . . . . 9 | |
12 | 9, 10, 11 | 3eqtr3g 2679 | . . . . . . . 8 |
13 | 12 | ex 450 | . . . . . . 7 |
14 | id 22 | . . . . . . . . . 10 | |
15 | 14 | sqxpeqd 5141 | . . . . . . . . 9 |
16 | 15 | fneq2d 5982 | . . . . . . . 8 |
17 | 4, 16 | syl5ibrcom 237 | . . . . . . 7 |
18 | 13, 17 | impbid 202 | . . . . . 6 |
19 | 18 | anbi1d 741 | . . . . 5 |
20 | 19 | exbidv 1850 | . . . 4 |
21 | 1, 20 | syl5bb 272 | . . 3 cat |
22 | isssc.3 | . . . 4 | |
23 | pweq 4161 | . . . . . 6 | |
24 | 23 | rexeqdv 3145 | . . . . 5 |
25 | 24 | ceqsexgv 3335 | . . . 4 |
26 | 22, 25 | syl 17 | . . 3 |
27 | 21, 26 | bitrd 268 | . 2 cat |
28 | df-rex 2918 | . . 3 | |
29 | 3anass 1042 | . . . . . . . 8 | |
30 | elixp2 7912 | . . . . . . . 8 | |
31 | vex 3203 | . . . . . . . . . . . 12 | |
32 | 31, 31 | xpex 6962 | . . . . . . . . . . 11 |
33 | fnex 6481 | . . . . . . . . . . 11 | |
34 | 32, 33 | mpan2 707 | . . . . . . . . . 10 |
35 | 34 | adantr 481 | . . . . . . . . 9 |
36 | 35 | pm4.71ri 665 | . . . . . . . 8 |
37 | 29, 30, 36 | 3bitr4i 292 | . . . . . . 7 |
38 | fndm 5990 | . . . . . . . . . . . . . 14 | |
39 | 38 | adantl 482 | . . . . . . . . . . . . 13 |
40 | isssc.1 | . . . . . . . . . . . . . . 15 | |
41 | 40 | adantr 481 | . . . . . . . . . . . . . 14 |
42 | fndm 5990 | . . . . . . . . . . . . . 14 | |
43 | 41, 42 | syl 17 | . . . . . . . . . . . . 13 |
44 | 39, 43 | eqtr3d 2658 | . . . . . . . . . . . 12 |
45 | 44 | dmeqd 5326 | . . . . . . . . . . 11 |
46 | dmxpid 5345 | . . . . . . . . . . 11 | |
47 | dmxpid 5345 | . . . . . . . . . . 11 | |
48 | 45, 46, 47 | 3eqtr3g 2679 | . . . . . . . . . 10 |
49 | 48 | ex 450 | . . . . . . . . 9 |
50 | id 22 | . . . . . . . . . . . 12 | |
51 | 50 | sqxpeqd 5141 | . . . . . . . . . . 11 |
52 | 51 | fneq2d 5982 | . . . . . . . . . 10 |
53 | 40, 52 | syl5ibrcom 237 | . . . . . . . . 9 |
54 | 49, 53 | impbid 202 | . . . . . . . 8 |
55 | 54 | anbi1d 741 | . . . . . . 7 |
56 | 37, 55 | syl5bb 272 | . . . . . 6 |
57 | 56 | anbi2d 740 | . . . . 5 |
58 | an12 838 | . . . . 5 | |
59 | 57, 58 | syl6bb 276 | . . . 4 |
60 | 59 | exbidv 1850 | . . 3 |
61 | 28, 60 | syl5bb 272 | . 2 |
62 | exsimpl 1795 | . . . . 5 | |
63 | isset 3207 | . . . . 5 | |
64 | 62, 63 | sylibr 224 | . . . 4 |
65 | 64 | a1i 11 | . . 3 |
66 | ssexg 4804 | . . . . . 6 | |
67 | 66 | expcom 451 | . . . . 5 |
68 | 22, 67 | syl 17 | . . . 4 |
69 | 68 | adantrd 484 | . . 3 |
70 | 31 | elpw 4164 | . . . . . . 7 |
71 | sseq1 3626 | . . . . . . 7 | |
72 | 70, 71 | syl5bb 272 | . . . . . 6 |
73 | 51 | raleqdv 3144 | . . . . . . 7 |
74 | fvex 6201 | . . . . . . . . . 10 | |
75 | 74 | elpw 4164 | . . . . . . . . 9 |
76 | fveq2 6191 | . . . . . . . . . . 11 | |
77 | df-ov 6653 | . . . . . . . . . . 11 | |
78 | 76, 77 | syl6eqr 2674 | . . . . . . . . . 10 |
79 | fveq2 6191 | . . . . . . . . . . 11 | |
80 | df-ov 6653 | . . . . . . . . . . 11 | |
81 | 79, 80 | syl6eqr 2674 | . . . . . . . . . 10 |
82 | 78, 81 | sseq12d 3634 | . . . . . . . . 9 |
83 | 75, 82 | syl5bb 272 | . . . . . . . 8 |
84 | 83 | ralxp 5263 | . . . . . . 7 |
85 | 73, 84 | syl6bb 276 | . . . . . 6 |
86 | 72, 85 | anbi12d 747 | . . . . 5 |
87 | 86 | ceqsexgv 3335 | . . . 4 |
88 | 87 | a1i 11 | . . 3 |
89 | 65, 69, 88 | pm5.21ndd 369 | . 2 |
90 | 27, 61, 89 | 3bitrd 294 | 1 cat |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wex 1704 wcel 1990 wral 2912 wrex 2913 cvv 3200 wss 3574 cpw 4158 cop 4183 class class class wbr 4653 cxp 5112 cdm 5114 wfn 5883 cfv 5888 (class class class)co 6650 cixp 7908 cat cssc 16467 |
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-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-ixp 7909 df-ssc 16470 |
This theorem is referenced by: ssc1 16481 ssc2 16482 sscres 16483 ssctr 16485 0ssc 16497 catsubcat 16499 rnghmsscmap2 41973 rnghmsscmap 41974 rhmsscmap2 42019 rhmsscmap 42020 rhmsscrnghm 42026 srhmsubc 42076 fldhmsubc 42084 srhmsubcALTV 42094 fldhmsubcALTV 42102 |
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