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Mirrors > Home > ILE Home > Th. List > f1ompt | Unicode version |
Description: Express bijection for a mapping operation. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by Mario Carneiro, 4-Dec-2016.) |
Ref | Expression |
---|---|
fmpt.1 |
Ref | Expression |
---|---|
f1ompt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 5267 | . . . . 5 | |
2 | dff1o4 5368 | . . . . . 6 | |
3 | 2 | baib 904 | . . . . 5 |
4 | 1, 3 | syl 14 | . . . 4 |
5 | fnres 5234 | . . . . . 6 | |
6 | nfcv 2279 | . . . . . . . . . 10 | |
7 | fmpt.1 | . . . . . . . . . . 11 | |
8 | nfmpt1 4016 | . . . . . . . . . . 11 | |
9 | 7, 8 | nfcxfr 2276 | . . . . . . . . . 10 |
10 | nfcv 2279 | . . . . . . . . . 10 | |
11 | 6, 9, 10 | nfbr 3969 | . . . . . . . . 9 |
12 | nfv 1508 | . . . . . . . . 9 | |
13 | breq1 3927 | . . . . . . . . . 10 | |
14 | df-mpt 3986 | . . . . . . . . . . . . 13 | |
15 | 7, 14 | eqtri 2158 | . . . . . . . . . . . 12 |
16 | 15 | breqi 3930 | . . . . . . . . . . 11 |
17 | df-br 3925 | . . . . . . . . . . . 12 | |
18 | opabid 4174 | . . . . . . . . . . . 12 | |
19 | 17, 18 | bitri 183 | . . . . . . . . . . 11 |
20 | 16, 19 | bitri 183 | . . . . . . . . . 10 |
21 | 13, 20 | syl6bb 195 | . . . . . . . . 9 |
22 | 11, 12, 21 | cbveu 2021 | . . . . . . . 8 |
23 | vex 2684 | . . . . . . . . . 10 | |
24 | vex 2684 | . . . . . . . . . 10 | |
25 | 23, 24 | brcnv 4717 | . . . . . . . . 9 |
26 | 25 | eubii 2006 | . . . . . . . 8 |
27 | df-reu 2421 | . . . . . . . 8 | |
28 | 22, 26, 27 | 3bitr4i 211 | . . . . . . 7 |
29 | 28 | ralbii 2439 | . . . . . 6 |
30 | 5, 29 | bitri 183 | . . . . 5 |
31 | relcnv 4912 | . . . . . . 7 | |
32 | df-rn 4545 | . . . . . . . 8 | |
33 | frn 5276 | . . . . . . . 8 | |
34 | 32, 33 | eqsstrrid 3139 | . . . . . . 7 |
35 | relssres 4852 | . . . . . . 7 | |
36 | 31, 34, 35 | sylancr 410 | . . . . . 6 |
37 | 36 | fneq1d 5208 | . . . . 5 |
38 | 30, 37 | syl5bbr 193 | . . . 4 |
39 | 4, 38 | bitr4d 190 | . . 3 |
40 | 39 | pm5.32i 449 | . 2 |
41 | f1of 5360 | . . 3 | |
42 | 41 | pm4.71ri 389 | . 2 |
43 | 7 | fmpt 5563 | . . 3 |
44 | 43 | anbi1i 453 | . 2 |
45 | 40, 42, 44 | 3bitr4i 211 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 wcel 1480 weu 1997 wral 2414 wreu 2416 wss 3066 cop 3525 class class class wbr 3924 copab 3983 cmpt 3984 ccnv 4533 cdm 4534 crn 4535 cres 4536 wrel 4539 wfn 5113 wf 5114 wf1o 5117 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 |
This theorem is referenced by: xpf1o 6731 icoshftf1o 9767 |
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