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Mirrors > Home > MPE Home > Th. List > fnun | Structured version Visualization version Unicode version |
Description: The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.) |
Ref | Expression |
---|---|
fnun |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fn 5891 | . . 3 | |
2 | df-fn 5891 | . . 3 | |
3 | ineq12 3809 | . . . . . . . . . . 11 | |
4 | 3 | eqeq1d 2624 | . . . . . . . . . 10 |
5 | 4 | anbi2d 740 | . . . . . . . . 9 |
6 | funun 5932 | . . . . . . . . 9 | |
7 | 5, 6 | syl6bir 244 | . . . . . . . 8 |
8 | dmun 5331 | . . . . . . . . 9 | |
9 | uneq12 3762 | . . . . . . . . 9 | |
10 | 8, 9 | syl5eq 2668 | . . . . . . . 8 |
11 | 7, 10 | jctird 567 | . . . . . . 7 |
12 | df-fn 5891 | . . . . . . 7 | |
13 | 11, 12 | syl6ibr 242 | . . . . . 6 |
14 | 13 | expd 452 | . . . . 5 |
15 | 14 | impcom 446 | . . . 4 |
16 | 15 | an4s 869 | . . 3 |
17 | 1, 2, 16 | syl2anb 496 | . 2 |
18 | 17 | imp 445 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 cun 3572 cin 3573 c0 3915 cdm 5114 wfun 5882 wfn 5883 |
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-fun 5890 df-fn 5891 |
This theorem is referenced by: fnunsn 5998 fun 6066 foun 6155 f1oun 6156 undifixp 7944 brwdom2 8478 sseqfn 30452 bnj927 30839 bnj535 30960 fullfunfnv 32053 finixpnum 33394 poimirlem1 33410 poimirlem2 33411 poimirlem3 33412 poimirlem4 33413 poimirlem6 33415 poimirlem7 33416 poimirlem11 33420 poimirlem12 33421 poimirlem16 33425 poimirlem17 33426 poimirlem19 33428 poimirlem20 33429 |
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