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Mirrors > Home > ILE Home > Th. List > resasplitss | Unicode version |
Description: If two functions agree on their common domain, their union contains a union of three functions with pairwise disjoint domains. If we assumed the law of the excluded middle, this would be equality rather than subset. (Contributed by Jim Kingdon, 28-Dec-2018.) |
Ref | Expression |
---|---|
resasplitss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unidm 3086 | . . . 4 | |
2 | 1 | uneq1i 3093 | . . 3 |
3 | un4 3103 | . . . 4 | |
4 | simp3 906 | . . . . . . 7 | |
5 | 4 | uneq1d 3096 | . . . . . 6 |
6 | 5 | uneq2d 3097 | . . . . 5 |
7 | resundi 4625 | . . . . . . 7 | |
8 | inundifss 3301 | . . . . . . . 8 | |
9 | ssres2 4638 | . . . . . . . 8 | |
10 | 8, 9 | ax-mp 7 | . . . . . . 7 |
11 | 7, 10 | eqsstr3i 2976 | . . . . . 6 |
12 | resundi 4625 | . . . . . . 7 | |
13 | incom 3129 | . . . . . . . . . 10 | |
14 | 13 | uneq1i 3093 | . . . . . . . . 9 |
15 | inundifss 3301 | . . . . . . . . 9 | |
16 | 14, 15 | eqsstri 2975 | . . . . . . . 8 |
17 | ssres2 4638 | . . . . . . . 8 | |
18 | 16, 17 | ax-mp 7 | . . . . . . 7 |
19 | 12, 18 | eqsstr3i 2976 | . . . . . 6 |
20 | unss12 3115 | . . . . . 6 | |
21 | 11, 19, 20 | mp2an 402 | . . . . 5 |
22 | 6, 21 | syl6eqss 2995 | . . . 4 |
23 | 3, 22 | syl5eqssr 2990 | . . 3 |
24 | 2, 23 | syl5eqssr 2990 | . 2 |
25 | fnresdm 5008 | . . . 4 | |
26 | fnresdm 5008 | . . . 4 | |
27 | uneq12 3092 | . . . 4 | |
28 | 25, 26, 27 | syl2an 273 | . . 3 |
29 | 28 | 3adant3 924 | . 2 |
30 | 24, 29 | sseqtrd 2981 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 w3a 885 wceq 1243 cdif 2914 cun 2915 cin 2916 wss 2917 cres 4347 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-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-dm 4355 df-res 4357 df-fun 4904 df-fn 4905 |
This theorem is referenced by: (None) |
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