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Mirrors > Home > MPE Home > Th. List > fvco2 | Structured version Visualization version Unicode version |
Description: Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47. (Contributed by NM, 9-Oct-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 16-Oct-2014.) |
Ref | Expression |
---|---|
fvco2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnsnfv 5947 |
. . . . . 6
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2 | 1 | imaeq2d 5186 |
. . . . 5
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3 | imaco 5358 |
. . . . 5
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4 | 2, 3 | syl6reqr 2514 |
. . . 4
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5 | 4 | eleq2d 2524 |
. . 3
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6 | 5 | iotabidv 5585 |
. 2
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7 | dffv3 5883 |
. 2
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8 | dffv3 5883 |
. 2
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9 | 6, 7, 8 | 3eqtr4g 2520 |
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 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-8 1899 ax-9 1906 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 ax-sep 4538 ax-nul 4547 ax-pow 4594 ax-pr 4652 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1457 df-ex 1674 df-nf 1678 df-sb 1808 df-eu 2313 df-mo 2314 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ne 2634 df-ral 2753 df-rex 2754 df-rab 2757 df-v 3058 df-sbc 3279 df-dif 3418 df-un 3420 df-in 3422 df-ss 3429 df-nul 3743 df-if 3893 df-sn 3980 df-pr 3982 df-op 3986 df-uni 4212 df-br 4416 df-opab 4475 df-id 4767 df-xp 4858 df-rel 4859 df-cnv 4860 df-co 4861 df-dm 4862 df-rn 4863 df-res 4864 df-ima 4865 df-iota 5564 df-fun 5602 df-fn 5603 df-fv 5608 |
This theorem is referenced by: fvco 5963 fvco3 5964 fvco4i 5965 fvcofneq 6052 ofco 6577 curry1 6914 curry2 6917 enfixsn 7706 smobeth 9036 fpwwe 9096 addpqnq 9388 mulpqnq 9391 revco 12967 ccatco 12968 cshco 12969 swrdco 12970 isoval 15718 prdsidlem 16616 gsumwmhm 16677 prdsinvlem 16842 gsmsymgrfixlem1 17116 f1omvdconj 17135 pmtrfinv 17150 symggen 17159 symgtrinv 17161 pmtr3ncomlem1 17162 ringidval 17785 prdsmgp 17886 lmhmco 18314 evlslem1 18786 evlsvar 18794 chrrhm 19150 zrhcofipsgn 19209 dsmmbas2 19348 dsmm0cl 19351 frlmbas 19366 frlmup3 19406 frlmup4 19407 f1lindf 19428 lindfmm 19433 m1detdiag 19670 1stccnp 20525 prdstopn 20691 xpstopnlem2 20874 uniioombllem6 22594 0vfval 26273 cnre2csqlem 28764 mblfinlem2 32022 rabren3dioph 35702 hausgraph 36133 stoweidlem59 37957 afvco2 38715 |
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