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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 5450 | . . . 4 | |
2 | dff1o4 5450 | . . . 4 | |
3 | fnun 5304 | . . . . . . 7 | |
4 | 3 | ex 114 | . . . . . 6 |
5 | fnun 5304 | . . . . . . . 8 | |
6 | cnvun 5016 | . . . . . . . . 9 | |
7 | 6 | fneq1i 5292 | . . . . . . . 8 |
8 | 5, 7 | sylibr 133 | . . . . . . 7 |
9 | 8 | ex 114 | . . . . . 6 |
10 | 4, 9 | im2anan9 593 | . . . . 5 |
11 | 10 | an4s 583 | . . . 4 |
12 | 1, 2, 11 | syl2anb 289 | . . 3 |
13 | dff1o4 5450 | . . 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 1348 cun 3119 cin 3120 c0 3414 ccnv 4610 wfn 5193 wf1o 5197 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-id 4278 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 |
This theorem is referenced by: f1oprg 5486 unen 6794 zfz1isolem1 10775 ennnfonelemhf1o 12368 |
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