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Mirrors > Home > ILE Home > Th. List > fnmpoovd | Unicode version |
Description: A function with a Cartesian product as domain is a mapping with two arguments defined by its operation values. (Contributed by AV, 20-Feb-2019.) (Revised by AV, 3-Jul-2022.) |
Ref | Expression |
---|---|
fnmpoovd.m | |
fnmpoovd.s | |
fnmpoovd.d | |
fnmpoovd.c |
Ref | Expression |
---|---|
fnmpoovd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnmpoovd.m | . . 3 | |
2 | fnmpoovd.c | . . . . . 6 | |
3 | 2 | 3expb 1182 | . . . . 5 |
4 | 3 | ralrimivva 2514 | . . . 4 |
5 | eqid 2139 | . . . . 5 | |
6 | 5 | fnmpo 6100 | . . . 4 |
7 | 4, 6 | syl 14 | . . 3 |
8 | eqfnov2 5878 | . . 3 | |
9 | 1, 7, 8 | syl2anc 408 | . 2 |
10 | nfcv 2281 | . . . . . . . 8 | |
11 | nfcv 2281 | . . . . . . . 8 | |
12 | nfcv 2281 | . . . . . . . 8 | |
13 | nfcv 2281 | . . . . . . . 8 | |
14 | fnmpoovd.s | . . . . . . . 8 | |
15 | 10, 11, 12, 13, 14 | cbvmpo 5850 | . . . . . . 7 |
16 | 15 | eqcomi 2143 | . . . . . 6 |
17 | 16 | a1i 9 | . . . . 5 |
18 | 17 | oveqd 5791 | . . . 4 |
19 | 18 | eqeq2d 2151 | . . 3 |
20 | 19 | 2ralbidv 2459 | . 2 |
21 | simprl 520 | . . . . 5 | |
22 | simprr 521 | . . . . 5 | |
23 | fnmpoovd.d | . . . . . 6 | |
24 | 23 | 3expb 1182 | . . . . 5 |
25 | eqid 2139 | . . . . . 6 | |
26 | 25 | ovmpt4g 5893 | . . . . 5 |
27 | 21, 22, 24, 26 | syl3anc 1216 | . . . 4 |
28 | 27 | eqeq2d 2151 | . . 3 |
29 | 28 | 2ralbidva 2457 | . 2 |
30 | 9, 20, 29 | 3bitrd 213 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wral 2416 cxp 4537 wfn 5118 (class class class)co 5774 cmpo 5776 |
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-in1 603 ax-in2 604 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 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 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-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 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-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 |
This theorem is referenced by: (None) |
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