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Mirrors > Home > ILE Home > Th. List > fmpt2x | Unicode version |
Description: Functionality, domain and codomain of a class given by the "maps to" notation, where is not constant but depends on . (Contributed by NM, 29-Dec-2014.) |
Ref | Expression |
---|---|
fmpt2x.1 |
Ref | Expression |
---|---|
fmpt2x |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2560 | . . . . . . . 8 | |
2 | vex 2560 | . . . . . . . 8 | |
3 | 1, 2 | op1std 5775 | . . . . . . 7 |
4 | 3 | csbeq1d 2858 | . . . . . 6 |
5 | 1, 2 | op2ndd 5776 | . . . . . . . 8 |
6 | 5 | csbeq1d 2858 | . . . . . . 7 |
7 | 6 | csbeq2dv 2875 | . . . . . 6 |
8 | 4, 7 | eqtrd 2072 | . . . . 5 |
9 | 8 | eleq1d 2106 | . . . 4 |
10 | 9 | raliunxp 4477 | . . 3 |
11 | nfv 1421 | . . . . . . 7 | |
12 | nfv 1421 | . . . . . . 7 | |
13 | nfv 1421 | . . . . . . . . 9 | |
14 | nfcsb1v 2882 | . . . . . . . . . 10 | |
15 | 14 | nfcri 2172 | . . . . . . . . 9 |
16 | 13, 15 | nfan 1457 | . . . . . . . 8 |
17 | nfcsb1v 2882 | . . . . . . . . 9 | |
18 | 17 | nfeq2 2189 | . . . . . . . 8 |
19 | 16, 18 | nfan 1457 | . . . . . . 7 |
20 | nfv 1421 | . . . . . . . 8 | |
21 | nfcv 2178 | . . . . . . . . . 10 | |
22 | nfcsb1v 2882 | . . . . . . . . . 10 | |
23 | 21, 22 | nfcsb 2884 | . . . . . . . . 9 |
24 | 23 | nfeq2 2189 | . . . . . . . 8 |
25 | 20, 24 | nfan 1457 | . . . . . . 7 |
26 | eleq1 2100 | . . . . . . . . . 10 | |
27 | 26 | adantr 261 | . . . . . . . . 9 |
28 | eleq1 2100 | . . . . . . . . . 10 | |
29 | csbeq1a 2860 | . . . . . . . . . . 11 | |
30 | 29 | eleq2d 2107 | . . . . . . . . . 10 |
31 | 28, 30 | sylan9bbr 436 | . . . . . . . . 9 |
32 | 27, 31 | anbi12d 442 | . . . . . . . 8 |
33 | csbeq1a 2860 | . . . . . . . . . 10 | |
34 | csbeq1a 2860 | . . . . . . . . . 10 | |
35 | 33, 34 | sylan9eqr 2094 | . . . . . . . . 9 |
36 | 35 | eqeq2d 2051 | . . . . . . . 8 |
37 | 32, 36 | anbi12d 442 | . . . . . . 7 |
38 | 11, 12, 19, 25, 37 | cbvoprab12 5578 | . . . . . 6 |
39 | df-mpt2 5517 | . . . . . 6 | |
40 | df-mpt2 5517 | . . . . . 6 | |
41 | 38, 39, 40 | 3eqtr4i 2070 | . . . . 5 |
42 | fmpt2x.1 | . . . . 5 | |
43 | 8 | mpt2mptx 5595 | . . . . 5 |
44 | 41, 42, 43 | 3eqtr4i 2070 | . . . 4 |
45 | 44 | fmpt 5319 | . . 3 |
46 | 10, 45 | bitr3i 175 | . 2 |
47 | nfv 1421 | . . 3 | |
48 | 17 | nfel1 2188 | . . . 4 |
49 | 14, 48 | nfralxy 2360 | . . 3 |
50 | nfv 1421 | . . . . 5 | |
51 | 22 | nfel1 2188 | . . . . 5 |
52 | 33 | eleq1d 2106 | . . . . 5 |
53 | 50, 51, 52 | cbvral 2529 | . . . 4 |
54 | 34 | eleq1d 2106 | . . . . 5 |
55 | 29, 54 | raleqbidv 2517 | . . . 4 |
56 | 53, 55 | syl5bb 181 | . . 3 |
57 | 47, 49, 56 | cbvral 2529 | . 2 |
58 | nfcv 2178 | . . . 4 | |
59 | nfcv 2178 | . . . . 5 | |
60 | 59, 14 | nfxp 4371 | . . . 4 |
61 | sneq 3386 | . . . . 5 | |
62 | 61, 29 | xpeq12d 4370 | . . . 4 |
63 | 58, 60, 62 | cbviun 3694 | . . 3 |
64 | 63 | feq2i 5040 | . 2 |
65 | 46, 57, 64 | 3bitr4i 201 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wb 98 wceq 1243 wcel 1393 wral 2306 csb 2852 csn 3375 cop 3378 ciun 3657 cmpt 3818 cxp 4343 wf 4898 cfv 4902 coprab 5513 cmpt2 5514 c1st 5765 c2nd 5766 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fv 4910 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 |
This theorem is referenced by: fmpt2 5827 |
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