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Mirrors > Home > MPE Home > Th. List > fun | Structured version Visualization version Unicode version |
Description: The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.) |
Ref | Expression |
---|---|
fun |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnun 5997 | . . . . 5 | |
2 | 1 | expcom 451 | . . . 4 |
3 | rnun 5541 | . . . . . 6 | |
4 | unss12 3785 | . . . . . 6 | |
5 | 3, 4 | syl5eqss 3649 | . . . . 5 |
6 | 5 | a1i 11 | . . . 4 |
7 | 2, 6 | anim12d 586 | . . 3 |
8 | df-f 5892 | . . . . 5 | |
9 | df-f 5892 | . . . . 5 | |
10 | 8, 9 | anbi12i 733 | . . . 4 |
11 | an4 865 | . . . 4 | |
12 | 10, 11 | bitri 264 | . . 3 |
13 | df-f 5892 | . . 3 | |
14 | 7, 12, 13 | 3imtr4g 285 | . 2 |
15 | 14 | impcom 446 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 cun 3572 cin 3573 wss 3574 c0 3915 crn 5115 wfn 5883 wf 5884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-fun 5890 df-fn 5891 df-f 5892 |
This theorem is referenced by: fun2 6067 ftpg 6423 fsnunf 6451 ralxpmap 7907 hashfxnn0 13124 hashfOLD 13126 cats1un 13475 pwssplit1 19059 axlowdimlem10 25831 wlkp1 26578 padct 29497 eulerpartlemt 30433 sseqf 30454 poimirlem3 33412 poimirlem16 33425 poimirlem19 33428 poimirlem22 33431 poimirlem23 33432 poimirlem24 33433 poimirlem25 33434 poimirlem28 33437 poimirlem29 33438 poimirlem31 33440 mapfzcons 37279 diophrw 37322 diophren 37377 pwssplit4 37659 aacllem 42547 |
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