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Mirrors > Home > MPE Home > Th. List > ciclcl | Structured version Visualization version GIF version |
Description: Isomorphism implies the left side is an object. (Contributed by AV, 5-Apr-2020.) |
Ref | Expression |
---|---|
ciclcl | β’ ((πΆ β Cat β§ π ( βπ βπΆ)π) β π β (BaseβπΆ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cicfval 17751 | . . . 4 β’ (πΆ β Cat β ( βπ βπΆ) = ((IsoβπΆ) supp β )) | |
2 | 1 | breqd 5152 | . . 3 β’ (πΆ β Cat β (π ( βπ βπΆ)π β π ((IsoβπΆ) supp β )π)) |
3 | isofn 17729 | . . . . 5 β’ (πΆ β Cat β (IsoβπΆ) Fn ((BaseβπΆ) Γ (BaseβπΆ))) | |
4 | fvexd 6899 | . . . . 5 β’ (πΆ β Cat β (IsoβπΆ) β V) | |
5 | 0ex 5300 | . . . . . 6 β’ β β V | |
6 | 5 | a1i 11 | . . . . 5 β’ (πΆ β Cat β β β V) |
7 | df-br 5142 | . . . . . 6 β’ (π ((IsoβπΆ) supp β )π β β¨π , πβ© β ((IsoβπΆ) supp β )) | |
8 | elsuppfng 8152 | . . . . . 6 β’ (((IsoβπΆ) Fn ((BaseβπΆ) Γ (BaseβπΆ)) β§ (IsoβπΆ) β V β§ β β V) β (β¨π , πβ© β ((IsoβπΆ) supp β ) β (β¨π , πβ© β ((BaseβπΆ) Γ (BaseβπΆ)) β§ ((IsoβπΆ)ββ¨π , πβ©) β β ))) | |
9 | 7, 8 | bitrid 283 | . . . . 5 β’ (((IsoβπΆ) Fn ((BaseβπΆ) Γ (BaseβπΆ)) β§ (IsoβπΆ) β V β§ β β V) β (π ((IsoβπΆ) supp β )π β (β¨π , πβ© β ((BaseβπΆ) Γ (BaseβπΆ)) β§ ((IsoβπΆ)ββ¨π , πβ©) β β ))) |
10 | 3, 4, 6, 9 | syl3anc 1368 | . . . 4 β’ (πΆ β Cat β (π ((IsoβπΆ) supp β )π β (β¨π , πβ© β ((BaseβπΆ) Γ (BaseβπΆ)) β§ ((IsoβπΆ)ββ¨π , πβ©) β β ))) |
11 | opelxp1 5711 | . . . . 5 β’ (β¨π , πβ© β ((BaseβπΆ) Γ (BaseβπΆ)) β π β (BaseβπΆ)) | |
12 | 11 | adantr 480 | . . . 4 β’ ((β¨π , πβ© β ((BaseβπΆ) Γ (BaseβπΆ)) β§ ((IsoβπΆ)ββ¨π , πβ©) β β ) β π β (BaseβπΆ)) |
13 | 10, 12 | biimtrdi 252 | . . 3 β’ (πΆ β Cat β (π ((IsoβπΆ) supp β )π β π β (BaseβπΆ))) |
14 | 2, 13 | sylbid 239 | . 2 β’ (πΆ β Cat β (π ( βπ βπΆ)π β π β (BaseβπΆ))) |
15 | 14 | imp 406 | 1 β’ ((πΆ β Cat β§ π ( βπ βπΆ)π) β π β (BaseβπΆ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 β wcel 2098 β wne 2934 Vcvv 3468 β c0 4317 β¨cop 4629 class class class wbr 5141 Γ cxp 5667 Fn wfn 6531 βcfv 6536 (class class class)co 7404 supp csupp 8143 Basecbs 17151 Catccat 17615 Isociso 17700 βπ ccic 17749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-supp 8144 df-inv 17702 df-iso 17703 df-cic 17750 |
This theorem is referenced by: cicsym 17758 cictr 17759 cicer 17760 initoeu2 17976 |
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