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Mirrors > Home > ILE Home > Th. List > fmpox | 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 |
---|---|
fmpox.1 |
Ref | Expression |
---|---|
fmpox |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2689 | . . . . . . . 8 | |
2 | vex 2689 | . . . . . . . 8 | |
3 | 1, 2 | op1std 6046 | . . . . . . 7 |
4 | 3 | csbeq1d 3010 | . . . . . 6 |
5 | 1, 2 | op2ndd 6047 | . . . . . . . 8 |
6 | 5 | csbeq1d 3010 | . . . . . . 7 |
7 | 6 | csbeq2dv 3028 | . . . . . 6 |
8 | 4, 7 | eqtrd 2172 | . . . . 5 |
9 | 8 | eleq1d 2208 | . . . 4 |
10 | 9 | raliunxp 4680 | . . 3 |
11 | nfv 1508 | . . . . . . 7 | |
12 | nfv 1508 | . . . . . . 7 | |
13 | nfv 1508 | . . . . . . . . 9 | |
14 | nfcsb1v 3035 | . . . . . . . . . 10 | |
15 | 14 | nfcri 2275 | . . . . . . . . 9 |
16 | 13, 15 | nfan 1544 | . . . . . . . 8 |
17 | nfcsb1v 3035 | . . . . . . . . 9 | |
18 | 17 | nfeq2 2293 | . . . . . . . 8 |
19 | 16, 18 | nfan 1544 | . . . . . . 7 |
20 | nfv 1508 | . . . . . . . 8 | |
21 | nfcv 2281 | . . . . . . . . . 10 | |
22 | nfcsb1v 3035 | . . . . . . . . . 10 | |
23 | 21, 22 | nfcsb 3037 | . . . . . . . . 9 |
24 | 23 | nfeq2 2293 | . . . . . . . 8 |
25 | 20, 24 | nfan 1544 | . . . . . . 7 |
26 | eleq1 2202 | . . . . . . . . . 10 | |
27 | 26 | adantr 274 | . . . . . . . . 9 |
28 | eleq1 2202 | . . . . . . . . . 10 | |
29 | csbeq1a 3012 | . . . . . . . . . . 11 | |
30 | 29 | eleq2d 2209 | . . . . . . . . . 10 |
31 | 28, 30 | sylan9bbr 458 | . . . . . . . . 9 |
32 | 27, 31 | anbi12d 464 | . . . . . . . 8 |
33 | csbeq1a 3012 | . . . . . . . . . 10 | |
34 | csbeq1a 3012 | . . . . . . . . . 10 | |
35 | 33, 34 | sylan9eqr 2194 | . . . . . . . . 9 |
36 | 35 | eqeq2d 2151 | . . . . . . . 8 |
37 | 32, 36 | anbi12d 464 | . . . . . . 7 |
38 | 11, 12, 19, 25, 37 | cbvoprab12 5845 | . . . . . 6 |
39 | df-mpo 5779 | . . . . . 6 | |
40 | df-mpo 5779 | . . . . . 6 | |
41 | 38, 39, 40 | 3eqtr4i 2170 | . . . . 5 |
42 | fmpox.1 | . . . . 5 | |
43 | 8 | mpomptx 5862 | . . . . 5 |
44 | 41, 42, 43 | 3eqtr4i 2170 | . . . 4 |
45 | 44 | fmpt 5570 | . . 3 |
46 | 10, 45 | bitr3i 185 | . 2 |
47 | nfv 1508 | . . 3 | |
48 | 17 | nfel1 2292 | . . . 4 |
49 | 14, 48 | nfralxy 2471 | . . 3 |
50 | nfv 1508 | . . . . 5 | |
51 | 22 | nfel1 2292 | . . . . 5 |
52 | 33 | eleq1d 2208 | . . . . 5 |
53 | 50, 51, 52 | cbvral 2650 | . . . 4 |
54 | 34 | eleq1d 2208 | . . . . 5 |
55 | 29, 54 | raleqbidv 2638 | . . . 4 |
56 | 53, 55 | syl5bb 191 | . . 3 |
57 | 47, 49, 56 | cbvral 2650 | . 2 |
58 | nfcv 2281 | . . . 4 | |
59 | nfcv 2281 | . . . . 5 | |
60 | 59, 14 | nfxp 4566 | . . . 4 |
61 | sneq 3538 | . . . . 5 | |
62 | 61, 29 | xpeq12d 4564 | . . . 4 |
63 | 58, 60, 62 | cbviun 3850 | . . 3 |
64 | 63 | feq2i 5266 | . 2 |
65 | 46, 57, 64 | 3bitr4i 211 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 csb 3003 csn 3527 cop 3530 ciun 3813 cmpt 3989 cxp 4537 wf 5119 cfv 5123 coprab 5775 cmpo 5776 c1st 6036 c2nd 6037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 |
This theorem is referenced by: fmpo 6099 |
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