Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cicfval | Structured version Visualization version GIF version |
Description: The set of isomorphic objects of the category π. (Contributed by AV, 4-Apr-2020.) |
Ref | Expression |
---|---|
cicfval | β’ (πΆ β Cat β ( βπ βπΆ) = ((IsoβπΆ) supp β )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cic 17605 | . 2 β’ βπ = (π β Cat β¦ ((Isoβπ) supp β )) | |
2 | fveq2 6825 | . . 3 β’ (π = πΆ β (Isoβπ) = (IsoβπΆ)) | |
3 | 2 | oveq1d 7352 | . 2 β’ (π = πΆ β ((Isoβπ) supp β ) = ((IsoβπΆ) supp β )) |
4 | id 22 | . 2 β’ (πΆ β Cat β πΆ β Cat) | |
5 | ovexd 7372 | . 2 β’ (πΆ β Cat β ((IsoβπΆ) supp β ) β V) | |
6 | 1, 3, 4, 5 | fvmptd3 6954 | 1 β’ (πΆ β Cat β ( βπ βπΆ) = ((IsoβπΆ) supp β )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1540 β wcel 2105 Vcvv 3441 β c0 4269 βcfv 6479 (class class class)co 7337 supp csupp 8047 Catccat 17470 Isociso 17555 βπ ccic 17604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pr 5372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-iota 6431 df-fun 6481 df-fv 6487 df-ov 7340 df-cic 17605 |
This theorem is referenced by: brcic 17607 ciclcl 17611 cicrcl 17612 cicer 17615 |
Copyright terms: Public domain | W3C validator |