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Mirrors > Home > ILE Home > Th. List > foun | Unicode version |
Description: The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.) |
Ref | Expression |
---|---|
foun |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fofn 5394 | . . . 4 | |
2 | fofn 5394 | . . . 4 | |
3 | 1, 2 | anim12i 336 | . . 3 |
4 | fnun 5276 | . . 3 | |
5 | 3, 4 | sylan 281 | . 2 |
6 | rnun 4994 | . . 3 | |
7 | forn 5395 | . . . . 5 | |
8 | 7 | ad2antrr 480 | . . . 4 |
9 | forn 5395 | . . . . 5 | |
10 | 9 | ad2antlr 481 | . . . 4 |
11 | 8, 10 | uneq12d 3262 | . . 3 |
12 | 6, 11 | syl5eq 2202 | . 2 |
13 | df-fo 5176 | . 2 | |
14 | 5, 12, 13 | sylanbrc 414 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 cun 3100 cin 3101 c0 3394 crn 4587 wfn 5165 wfo 5168 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-br 3966 df-opab 4026 df-id 4253 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-fun 5172 df-fn 5173 df-f 5174 df-fo 5176 |
This theorem is referenced by: (None) |
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