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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 2724 | . . . . . . . 8 | |
2 | vex 2724 | . . . . . . . 8 | |
3 | 1, 2 | op1std 6108 | . . . . . . 7 |
4 | 3 | csbeq1d 3047 | . . . . . 6 |
5 | 1, 2 | op2ndd 6109 | . . . . . . . 8 |
6 | 5 | csbeq1d 3047 | . . . . . . 7 |
7 | 6 | csbeq2dv 3066 | . . . . . 6 |
8 | 4, 7 | eqtrd 2197 | . . . . 5 |
9 | 8 | eleq1d 2233 | . . . 4 |
10 | 9 | raliunxp 4739 | . . 3 |
11 | nfv 1515 | . . . . . . 7 | |
12 | nfv 1515 | . . . . . . 7 | |
13 | nfv 1515 | . . . . . . . . 9 | |
14 | nfcsb1v 3073 | . . . . . . . . . 10 | |
15 | 14 | nfcri 2300 | . . . . . . . . 9 |
16 | 13, 15 | nfan 1552 | . . . . . . . 8 |
17 | nfcsb1v 3073 | . . . . . . . . 9 | |
18 | 17 | nfeq2 2318 | . . . . . . . 8 |
19 | 16, 18 | nfan 1552 | . . . . . . 7 |
20 | nfv 1515 | . . . . . . . 8 | |
21 | nfcv 2306 | . . . . . . . . . 10 | |
22 | nfcsb1v 3073 | . . . . . . . . . 10 | |
23 | 21, 22 | nfcsb 3077 | . . . . . . . . 9 |
24 | 23 | nfeq2 2318 | . . . . . . . 8 |
25 | 20, 24 | nfan 1552 | . . . . . . 7 |
26 | eleq1 2227 | . . . . . . . . . 10 | |
27 | 26 | adantr 274 | . . . . . . . . 9 |
28 | eleq1 2227 | . . . . . . . . . 10 | |
29 | csbeq1a 3049 | . . . . . . . . . . 11 | |
30 | 29 | eleq2d 2234 | . . . . . . . . . 10 |
31 | 28, 30 | sylan9bbr 459 | . . . . . . . . 9 |
32 | 27, 31 | anbi12d 465 | . . . . . . . 8 |
33 | csbeq1a 3049 | . . . . . . . . . 10 | |
34 | csbeq1a 3049 | . . . . . . . . . 10 | |
35 | 33, 34 | sylan9eqr 2219 | . . . . . . . . 9 |
36 | 35 | eqeq2d 2176 | . . . . . . . 8 |
37 | 32, 36 | anbi12d 465 | . . . . . . 7 |
38 | 11, 12, 19, 25, 37 | cbvoprab12 5907 | . . . . . 6 |
39 | df-mpo 5841 | . . . . . 6 | |
40 | df-mpo 5841 | . . . . . 6 | |
41 | 38, 39, 40 | 3eqtr4i 2195 | . . . . 5 |
42 | fmpox.1 | . . . . 5 | |
43 | 8 | mpomptx 5924 | . . . . 5 |
44 | 41, 42, 43 | 3eqtr4i 2195 | . . . 4 |
45 | 44 | fmpt 5629 | . . 3 |
46 | 10, 45 | bitr3i 185 | . 2 |
47 | nfv 1515 | . . 3 | |
48 | 17 | nfel1 2317 | . . . 4 |
49 | 14, 48 | nfralxy 2502 | . . 3 |
50 | nfv 1515 | . . . . 5 | |
51 | 22 | nfel1 2317 | . . . . 5 |
52 | 33 | eleq1d 2233 | . . . . 5 |
53 | 50, 51, 52 | cbvral 2685 | . . . 4 |
54 | 34 | eleq1d 2233 | . . . . 5 |
55 | 29, 54 | raleqbidv 2671 | . . . 4 |
56 | 53, 55 | syl5bb 191 | . . 3 |
57 | 47, 49, 56 | cbvral 2685 | . 2 |
58 | nfcv 2306 | . . . 4 | |
59 | nfcv 2306 | . . . . 5 | |
60 | 59, 14 | nfxp 4625 | . . . 4 |
61 | sneq 3581 | . . . . 5 | |
62 | 61, 29 | xpeq12d 4623 | . . . 4 |
63 | 58, 60, 62 | cbviun 3897 | . . 3 |
64 | 63 | feq2i 5325 | . 2 |
65 | 46, 57, 64 | 3bitr4i 211 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1342 wcel 2135 wral 2442 csb 3040 csn 3570 cop 3573 ciun 3860 cmpt 4037 cxp 4596 wf 5178 cfv 5182 coprab 5837 cmpo 5838 c1st 6098 c2nd 6099 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 |
This theorem is referenced by: fmpo 6161 |
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