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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 5321 | . . . . 5 | |
2 | dff1o4 5424 | . . . . . 6 | |
3 | 2 | baib 905 | . . . . 5 |
4 | 1, 3 | syl 14 | . . . 4 |
5 | fnres 5288 | . . . . . 6 | |
6 | nfcv 2299 | . . . . . . . . . 10 | |
7 | fmpt.1 | . . . . . . . . . . 11 | |
8 | nfmpt1 4059 | . . . . . . . . . . 11 | |
9 | 7, 8 | nfcxfr 2296 | . . . . . . . . . 10 |
10 | nfcv 2299 | . . . . . . . . . 10 | |
11 | 6, 9, 10 | nfbr 4012 | . . . . . . . . 9 |
12 | nfv 1508 | . . . . . . . . 9 | |
13 | breq1 3970 | . . . . . . . . . 10 | |
14 | df-mpt 4029 | . . . . . . . . . . . . 13 | |
15 | 7, 14 | eqtri 2178 | . . . . . . . . . . . 12 |
16 | 15 | breqi 3973 | . . . . . . . . . . 11 |
17 | df-br 3968 | . . . . . . . . . . . 12 | |
18 | opabid 4219 | . . . . . . . . . . . 12 | |
19 | 17, 18 | bitri 183 | . . . . . . . . . . 11 |
20 | 16, 19 | bitri 183 | . . . . . . . . . 10 |
21 | 13, 20 | bitrdi 195 | . . . . . . . . 9 |
22 | 11, 12, 21 | cbveu 2030 | . . . . . . . 8 |
23 | vex 2715 | . . . . . . . . . 10 | |
24 | vex 2715 | . . . . . . . . . 10 | |
25 | 23, 24 | brcnv 4771 | . . . . . . . . 9 |
26 | 25 | eubii 2015 | . . . . . . . 8 |
27 | df-reu 2442 | . . . . . . . 8 | |
28 | 22, 26, 27 | 3bitr4i 211 | . . . . . . 7 |
29 | 28 | ralbii 2463 | . . . . . 6 |
30 | 5, 29 | bitri 183 | . . . . 5 |
31 | relcnv 4966 | . . . . . . 7 | |
32 | df-rn 4599 | . . . . . . . 8 | |
33 | frn 5330 | . . . . . . . 8 | |
34 | 32, 33 | eqsstrrid 3175 | . . . . . . 7 |
35 | relssres 4906 | . . . . . . 7 | |
36 | 31, 34, 35 | sylancr 411 | . . . . . 6 |
37 | 36 | fneq1d 5262 | . . . . 5 |
38 | 30, 37 | bitr3id 193 | . . . 4 |
39 | 4, 38 | bitr4d 190 | . . 3 |
40 | 39 | pm5.32i 450 | . 2 |
41 | f1of 5416 | . . 3 | |
42 | 41 | pm4.71ri 390 | . 2 |
43 | 7 | fmpt 5619 | . . 3 |
44 | 43 | anbi1i 454 | . 2 |
45 | 40, 42, 44 | 3bitr4i 211 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1335 weu 2006 wcel 2128 wral 2435 wreu 2437 wss 3102 cop 3564 class class class wbr 3967 copab 4026 cmpt 4027 ccnv 4587 cdm 4588 crn 4589 cres 4590 wrel 4593 wfn 5167 wf 5168 wf1o 5171 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4028 df-mpt 4029 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 |
This theorem is referenced by: xpf1o 6791 icoshftf1o 9901 |
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