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Mirrors > Home > ILE Home > Th. List > fco | Unicode version |
Description: Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
fco |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f 5167 |
. . 3
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2 | df-f 5167 |
. . 3
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3 | fnco 5271 |
. . . . . . 7
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4 | 3 | 3expib 1185 |
. . . . . 6
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5 | 4 | adantr 274 |
. . . . 5
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6 | rncoss 4849 |
. . . . . . 7
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7 | sstr 3132 |
. . . . . . 7
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8 | 6, 7 | mpan 421 |
. . . . . 6
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9 | 8 | adantl 275 |
. . . . 5
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10 | 5, 9 | jctird 315 |
. . . 4
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11 | 10 | imp 123 |
. . 3
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12 | 1, 2, 11 | syl2anb 289 |
. 2
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13 | df-f 5167 |
. 2
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14 | 12, 13 | sylibr 133 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-14 2128 ax-ext 2136 ax-sep 4078 ax-pow 4130 ax-pr 4164 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ral 2437 df-rex 2438 df-v 2711 df-un 3102 df-in 3104 df-ss 3111 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-br 3962 df-opab 4022 df-id 4248 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-rn 4590 df-fun 5165 df-fn 5166 df-f 5167 |
This theorem is referenced by: fco2 5329 f1co 5380 foco 5395 mapen 6780 ctm 7039 enomnilem 7060 enmkvlem 7083 enwomnilem 7091 fnn0nninf 10314 fsumcl2lem 11272 fsumadd 11280 fprodmul 11465 algcvg 11896 cnco 12560 cnptopco 12561 lmtopcnp 12589 cnmpt11 12622 cnmpt21 12630 comet 12838 cnmet 12869 cncfco 12917 limccnpcntop 12983 dvcoapbr 13010 dvcjbr 13011 dvcj 13012 |
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