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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 3214 | . . . 4 | |
2 | 1 | uneq1i 3221 | . . 3 |
3 | un4 3231 | . . . 4 | |
4 | simp3 983 | . . . . . . 7 | |
5 | 4 | uneq1d 3224 | . . . . . 6 |
6 | 5 | uneq2d 3225 | . . . . 5 |
7 | resundi 4827 | . . . . . . 7 | |
8 | inundifss 3435 | . . . . . . . 8 | |
9 | ssres2 4841 | . . . . . . . 8 | |
10 | 8, 9 | ax-mp 5 | . . . . . . 7 |
11 | 7, 10 | eqsstrri 3125 | . . . . . 6 |
12 | resundi 4827 | . . . . . . 7 | |
13 | incom 3263 | . . . . . . . . . 10 | |
14 | 13 | uneq1i 3221 | . . . . . . . . 9 |
15 | inundifss 3435 | . . . . . . . . 9 | |
16 | 14, 15 | eqsstri 3124 | . . . . . . . 8 |
17 | ssres2 4841 | . . . . . . . 8 | |
18 | 16, 17 | ax-mp 5 | . . . . . . 7 |
19 | 12, 18 | eqsstrri 3125 | . . . . . 6 |
20 | unss12 3243 | . . . . . 6 | |
21 | 11, 19, 20 | mp2an 422 | . . . . 5 |
22 | 6, 21 | eqsstrdi 3144 | . . . 4 |
23 | 3, 22 | eqsstrrid 3139 | . . 3 |
24 | 2, 23 | eqsstrrid 3139 | . 2 |
25 | fnresdm 5227 | . . . 4 | |
26 | fnresdm 5227 | . . . 4 | |
27 | uneq12 3220 | . . . 4 | |
28 | 25, 26, 27 | syl2an 287 | . . 3 |
29 | 28 | 3adant3 1001 | . 2 |
30 | 24, 29 | sseqtrd 3130 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 w3a 962 wceq 1331 cdif 3063 cun 3064 cin 3065 wss 3066 cres 4536 wfn 5113 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-dm 4544 df-res 4546 df-fun 5120 df-fn 5121 |
This theorem is referenced by: (None) |
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