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Mirrors > Home > ILE Home > Th. List > fnun | 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 4905 | . . 3 | |
2 | df-fn 4905 | . . 3 | |
3 | ineq12 3133 | . . . . . . . . . . 11 | |
4 | 3 | eqeq1d 2048 | . . . . . . . . . 10 |
5 | 4 | anbi2d 437 | . . . . . . . . 9 |
6 | funun 4944 | . . . . . . . . 9 | |
7 | 5, 6 | syl6bir 153 | . . . . . . . 8 |
8 | dmun 4542 | . . . . . . . . 9 | |
9 | uneq12 3092 | . . . . . . . . 9 | |
10 | 8, 9 | syl5eq 2084 | . . . . . . . 8 |
11 | 7, 10 | jctird 300 | . . . . . . 7 |
12 | df-fn 4905 | . . . . . . 7 | |
13 | 11, 12 | syl6ibr 151 | . . . . . 6 |
14 | 13 | expd 245 | . . . . 5 |
15 | 14 | impcom 116 | . . . 4 |
16 | 15 | an4s 522 | . . 3 |
17 | 1, 2, 16 | syl2anb 275 | . 2 |
18 | 17 | imp 115 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 cun 2915 cin 2916 c0 3224 cdm 4345 wfun 4896 wfn 4897 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-fun 4904 df-fn 4905 |
This theorem is referenced by: fnunsn 5006 fun 5063 foun 5145 f1oun 5146 |
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