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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 1199 | . . . . 5 |
4 | 3 | ralrimivva 2552 | . . . 4 |
5 | eqid 2170 | . . . . 5 | |
6 | 5 | fnmpo 6178 | . . . 4 |
7 | 4, 6 | syl 14 | . . 3 |
8 | eqfnov2 5957 | . . 3 | |
9 | 1, 7, 8 | syl2anc 409 | . 2 |
10 | nfcv 2312 | . . . . . . . 8 | |
11 | nfcv 2312 | . . . . . . . 8 | |
12 | nfcv 2312 | . . . . . . . 8 | |
13 | nfcv 2312 | . . . . . . . 8 | |
14 | fnmpoovd.s | . . . . . . . 8 | |
15 | 10, 11, 12, 13, 14 | cbvmpo 5929 | . . . . . . 7 |
16 | 15 | eqcomi 2174 | . . . . . 6 |
17 | 16 | a1i 9 | . . . . 5 |
18 | 17 | oveqd 5867 | . . . 4 |
19 | 18 | eqeq2d 2182 | . . 3 |
20 | 19 | 2ralbidv 2494 | . 2 |
21 | simprl 526 | . . . . 5 | |
22 | simprr 527 | . . . . 5 | |
23 | fnmpoovd.d | . . . . . 6 | |
24 | 23 | 3expb 1199 | . . . . 5 |
25 | eqid 2170 | . . . . . 6 | |
26 | 25 | ovmpt4g 5972 | . . . . 5 |
27 | 21, 22, 24, 26 | syl3anc 1233 | . . . 4 |
28 | 27 | eqeq2d 2182 | . . 3 |
29 | 28 | 2ralbidva 2492 | . 2 |
30 | 9, 20, 29 | 3bitrd 213 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 973 wceq 1348 wcel 2141 wral 2448 cxp 4607 wfn 5191 (class class class)co 5850 cmpo 5852 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 df-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 |
This theorem is referenced by: (None) |
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