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Mirrors > Home > ILE Home > Th. List > foco | Unicode version |
Description: Composition of onto functions. (Contributed by NM, 22-Mar-2006.) |
Ref | Expression |
---|---|
foco |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffo2 5408 | . . 3 | |
2 | dffo2 5408 | . . 3 | |
3 | fco 5347 | . . . . 5 | |
4 | 3 | ad2ant2r 501 | . . . 4 |
5 | fdm 5337 | . . . . . . . 8 | |
6 | eqtr3 2184 | . . . . . . . 8 | |
7 | 5, 6 | sylan 281 | . . . . . . 7 |
8 | rncoeq 4871 | . . . . . . . . 9 | |
9 | 8 | eqeq1d 2173 | . . . . . . . 8 |
10 | 9 | biimpar 295 | . . . . . . 7 |
11 | 7, 10 | sylan 281 | . . . . . 6 |
12 | 11 | an32s 558 | . . . . 5 |
13 | 12 | adantrl 470 | . . . 4 |
14 | 4, 13 | jca 304 | . . 3 |
15 | 1, 2, 14 | syl2anb 289 | . 2 |
16 | dffo2 5408 | . 2 | |
17 | 15, 16 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 cdm 4598 crn 4599 ccom 4602 wf 5178 wfo 5180 |
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-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-fun 5184 df-fn 5185 df-f 5186 df-fo 5188 |
This theorem is referenced by: f1oco 5449 ennnfonelemnn0 12298 ctinfomlemom 12303 qnnen 12307 enctlem 12308 |
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