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Mirrors > Home > ILE Home > Th. List > casefun | Unicode version |
Description: The "case" construction of two functions is a function. (Contributed by BJ, 10-Jul-2022.) |
Ref | Expression |
---|---|
casefun.f | |
casefun.g |
Ref | Expression |
---|---|
casefun | case |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | casefun.f | . . . 4 | |
2 | djulf1o 6943 | . . . . . 6 inl | |
3 | f1of1 5366 | . . . . . 6 inl inl | |
4 | 2, 3 | ax-mp 5 | . . . . 5 inl |
5 | df-f1 5128 | . . . . . 6 inl inl inl | |
6 | 5 | simprbi 273 | . . . . 5 inl inl |
7 | 4, 6 | mp1i 10 | . . . 4 inl |
8 | funco 5163 | . . . 4 inl inl | |
9 | 1, 7, 8 | syl2anc 408 | . . 3 inl |
10 | casefun.g | . . . 4 | |
11 | djurf1o 6944 | . . . . . 6 inr | |
12 | f1of1 5366 | . . . . . 6 inr inr | |
13 | 11, 12 | ax-mp 5 | . . . . 5 inr |
14 | df-f1 5128 | . . . . . 6 inr inr inr | |
15 | 14 | simprbi 273 | . . . . 5 inr inr |
16 | 13, 15 | mp1i 10 | . . . 4 inr |
17 | funco 5163 | . . . 4 inr inr | |
18 | 10, 16, 17 | syl2anc 408 | . . 3 inr |
19 | dmcoss 4808 | . . . . . . 7 inl inl | |
20 | df-rn 4550 | . . . . . . 7 inl inl | |
21 | 19, 20 | sseqtrri 3132 | . . . . . 6 inl inl |
22 | dmcoss 4808 | . . . . . . 7 inr inr | |
23 | df-rn 4550 | . . . . . . 7 inr inr | |
24 | 22, 23 | sseqtrri 3132 | . . . . . 6 inr inr |
25 | ss2in 3304 | . . . . . 6 inl inl inr inr inl inr inl inr | |
26 | 21, 24, 25 | mp2an 422 | . . . . 5 inl inr inl inr |
27 | rnresv 4998 | . . . . . . . . 9 inl inl | |
28 | 27 | eqcomi 2143 | . . . . . . . 8 inl inl |
29 | rnresv 4998 | . . . . . . . . 9 inr inr | |
30 | 29 | eqcomi 2143 | . . . . . . . 8 inr inr |
31 | 28, 30 | ineq12i 3275 | . . . . . . 7 inl inr inl inr |
32 | djuinr 6948 | . . . . . . 7 inl inr | |
33 | 31, 32 | eqtri 2160 | . . . . . 6 inl inr |
34 | 33 | a1i 9 | . . . . 5 inl inr |
35 | 26, 34 | sseqtrid 3147 | . . . 4 inl inr |
36 | ss0 3403 | . . . 4 inl inr inl inr | |
37 | 35, 36 | syl 14 | . . 3 inl inr |
38 | funun 5167 | . . 3 inl inr inl inr inl inr | |
39 | 9, 18, 37, 38 | syl21anc 1215 | . 2 inl inr |
40 | df-case 6969 | . . 3 case inl inr | |
41 | 40 | funeqi 5144 | . 2 case inl inr |
42 | 39, 41 | sylibr 133 | 1 case |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 cvv 2686 cun 3069 cin 3070 wss 3071 c0 3363 csn 3527 cxp 4537 ccnv 4538 cdm 4539 crn 4540 cres 4541 ccom 4543 wfun 5117 wf 5119 wf1 5120 wf1o 5122 c1o 6306 inlcinl 6930 inrcinr 6931 casecdjucase 6968 |
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-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
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-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1st 6038 df-2nd 6039 df-1o 6313 df-inl 6932 df-inr 6933 df-case 6969 |
This theorem is referenced by: casef 6973 |
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