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Mirrors > Home > MPE Home > Th. List > f1oco | Structured version Visualization version Unicode version |
Description: Composition of one-to-one onto functions. (Contributed by NM, 19-Mar-1998.) |
Ref | Expression |
---|---|
f1oco |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f1o 5589 |
. . 3
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2 | df-f1o 5589 |
. . 3
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3 | f1co 5788 |
. . . . 5
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4 | foco 5803 |
. . . . 5
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5 | 3, 4 | anim12i 570 |
. . . 4
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6 | 5 | an4s 835 |
. . 3
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7 | 1, 2, 6 | syl2anb 482 |
. 2
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8 | df-f1o 5589 |
. 2
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9 | 7, 8 | sylibr 216 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pr 4639 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-rab 2746 df-v 3047 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-nul 3732 df-if 3882 df-sn 3969 df-pr 3971 df-op 3975 df-br 4403 df-opab 4462 df-id 4749 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 |
This theorem is referenced by: fveqf1o 6200 isotr 6227 ener 7616 omf1o 7675 enfixsn 7681 oef1o 8203 cnfcom3 8209 infxpenc 8449 ackbij2lem2 8670 canthp1lem2 9078 pwfseqlem5 9088 hashfacen 12617 summolem3 13780 fsumf1o 13789 ackbijnn 13886 prodmolem3 13987 fprodf1o 14000 eulerthlem2 14730 symgcl 17032 pmtrfconj 17107 gsumval3eu 17538 gsumval3lem1 17539 gsumval3 17541 lmimco 19402 resinf1o 23485 motco 24585 counop 27574 eulerpartgbij 29205 derangenlem 29894 subfacp1lem5 29907 poimirlem9 31949 poimirlem15 31955 poimirlem16 31956 poimirlem17 31957 poimirlem19 31959 poimirlem20 31960 rngoisoco 32221 lautco 33662 |
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