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Mirrors > Home > MPE Home > Th. List > f1oun | Structured version Visualization version 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 6145 | . . . 4 | |
2 | dff1o4 6145 | . . . 4 | |
3 | fnun 5997 | . . . . . . 7 | |
4 | 3 | ex 450 | . . . . . 6 |
5 | fnun 5997 | . . . . . . . 8 | |
6 | cnvun 5538 | . . . . . . . . 9 | |
7 | 6 | fneq1i 5985 | . . . . . . . 8 |
8 | 5, 7 | sylibr 224 | . . . . . . 7 |
9 | 8 | ex 450 | . . . . . 6 |
10 | 4, 9 | im2anan9 880 | . . . . 5 |
11 | 10 | an4s 869 | . . . 4 |
12 | 1, 2, 11 | syl2anb 496 | . . 3 |
13 | dff1o4 6145 | . . 3 | |
14 | 12, 13 | syl6ibr 242 | . 2 |
15 | 14 | imp 445 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 cun 3572 cin 3573 c0 3915 ccnv 5113 wfn 5883 wf1o 5887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 |
This theorem is referenced by: f1oprg 6181 fveqf1o 6557 oacomf1o 7645 unen 8040 enfixsn 8069 domss2 8119 isinf 8173 marypha1lem 8339 hashf1lem1 13239 f1oun2prg 13662 eupthp1 27076 isoun 29479 subfacp1lem2a 31162 subfacp1lem5 31166 poimirlem3 33412 poimirlem15 33424 poimirlem16 33425 poimirlem17 33426 poimirlem19 33428 poimirlem20 33429 eldioph2lem1 37323 eldioph2lem2 37324 |
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