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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 5343 | . . . 4 | |
2 | dff1o4 5343 | . . . 4 | |
3 | fnun 5199 | . . . . . . 7 | |
4 | 3 | ex 114 | . . . . . 6 |
5 | fnun 5199 | . . . . . . . 8 | |
6 | cnvun 4914 | . . . . . . . . 9 | |
7 | 6 | fneq1i 5187 | . . . . . . . 8 |
8 | 5, 7 | sylibr 133 | . . . . . . 7 |
9 | 8 | ex 114 | . . . . . 6 |
10 | 4, 9 | im2anan9 572 | . . . . 5 |
11 | 10 | an4s 562 | . . . 4 |
12 | 1, 2, 11 | syl2anb 289 | . . 3 |
13 | dff1o4 5343 | . . 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 1316 cun 3039 cin 3040 c0 3333 ccnv 4508 wfn 5088 wf1o 5092 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-id 4185 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 |
This theorem is referenced by: f1oprg 5379 unen 6678 zfz1isolem1 10551 ennnfonelemhf1o 11853 |
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