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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffo2 5442 |
. . 3
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2 | dffo2 5442 |
. . 3
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3 | fco 5381 |
. . . . 5
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4 | 3 | ad2ant2r 509 |
. . . 4
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5 | fdm 5371 |
. . . . . . . 8
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6 | eqtr3 2197 |
. . . . . . . 8
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7 | 5, 6 | sylan 283 |
. . . . . . 7
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8 | rncoeq 4900 |
. . . . . . . . 9
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9 | 8 | eqeq1d 2186 |
. . . . . . . 8
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10 | 9 | biimpar 297 |
. . . . . . 7
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11 | 7, 10 | sylan 283 |
. . . . . 6
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12 | 11 | an32s 568 |
. . . . 5
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13 | 12 | adantrl 478 |
. . . 4
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14 | 4, 13 | jca 306 |
. . 3
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15 | 1, 2, 14 | syl2anb 291 |
. 2
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16 | dffo2 5442 |
. 2
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17 | 15, 16 | sylibr 134 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-fun 5218 df-fn 5219 df-f 5220 df-fo 5222 |
This theorem is referenced by: f1oco 5484 ennnfonelemnn0 12417 ctinfomlemom 12422 qnnen 12426 enctlem 12427 |
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