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Mirrors > Home > ILE Home > Th. List > f1oun | Unicode version |
Description: The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.) |
Ref | Expression |
---|---|
f1oun |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dff1o4 5423 | . . . 4 | |
2 | dff1o4 5423 | . . . 4 | |
3 | fnun 5277 | . . . . . . 7 | |
4 | 3 | ex 114 | . . . . . 6 |
5 | fnun 5277 | . . . . . . . 8 | |
6 | cnvun 4992 | . . . . . . . . 9 | |
7 | 6 | fneq1i 5265 | . . . . . . . 8 |
8 | 5, 7 | sylibr 133 | . . . . . . 7 |
9 | 8 | ex 114 | . . . . . 6 |
10 | 4, 9 | im2anan9 588 | . . . . 5 |
11 | 10 | an4s 578 | . . . 4 |
12 | 1, 2, 11 | syl2anb 289 | . . 3 |
13 | dff1o4 5423 | . . 3 | |
14 | 12, 13 | syl6ibr 161 | . 2 |
15 | 14 | imp 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 cun 3100 cin 3101 c0 3394 ccnv 4586 wfn 5166 wf1o 5170 |
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 4083 ax-pow 4136 ax-pr 4170 |
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 3967 df-opab 4027 df-id 4254 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 |
This theorem is referenced by: f1oprg 5459 unen 6762 zfz1isolem1 10715 ennnfonelemhf1o 12184 |
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