Theorem idcvvidc 10759

Description: Functors preserve codomains. JFM CAT₁ th. 98.

Hypotheses

Ref	Expression
idcvvidc.1	|- M1 = dom (dom` T)
idcvvidc.2	|- C1 = (cod` T)
idcvvidc.3	|- I1 = (id` T)
idcvvidc.4	|- I2 = (id` U)
idcvvidc.5	|- C2 = (cod` U)

Assertion

Ref	Expression
idcvvidc	|- ((T e. Cat /\ U e. Cat) -> (F e. (Func` <.T, U>.) -> A.m e. M1 (F` (I1` (C1` m))) = (I2` (C2` (F` m)))))

Distinct variable groups:   m,F   m,M₁   T,m   U,m

Proof of Theorem idcvvidc

Step	Hyp	Ref	Expression
1		eqid 1475	. . . . . . . 8 |- dom (id` T) = dom (id` T)
2		idcvvidc.1	. . . . . . . 8 |- M1 = dom (dom` T)
3		eqid 1475	. . . . . . . 8 |- (dom` T) = (dom` T)
4		idcvvidc.2	. . . . . . . 8 |- C1 = (cod` T)
5		idcvvidc.3	. . . . . . . 8 |- I1 = (id` T)
6		eqid 1475	. . . . . . . 8 |- (o` T) = (o` T)
7		eqid 1475	. . . . . . . 8 |- dom (id` U) = dom (id` U)
8		eqid 1475	. . . . . . . 8 |- dom (dom` U) = dom (dom` U)
9		eqid 1475	. . . . . . . 8 |- (dom` U) = (dom` U)
10		idcvvidc.5	. . . . . . . 8 |- C2 = (cod` U)
11		idcvvidc.4	. . . . . . . 8 |- I2 = (id` U)
12		eqid 1475	. . . . . . . 8 |- (o` U) = (o` U)
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	isfunb 10755	. . . . . . 7 |- ((T e. Cat /\ U e. Cat /\ F e. (Func` <.T, U>.)) -> (F e. (Func` <.T, U>.) <-> (F:M1-->dom (dom` U) /\ (A.a e. dom (id` T)E.b e. dom (id` U)(F` (I1` a)) = (I2` b) /\ (A.m e. M1 (F` (I1` ((dom` T)` m))) = (I2` ((dom` U)` (F` m))) /\ A.m e. M1 (F` (I1` (C1` m))) = (I2` (C2` (F` m)))) /\ A.m e. M1 A.n e. M1 ((C1` n) = ((dom` T)` m) -> (F` (m(o` T)n)) = ((F` m)(o` U)(F` n)))))))
14	13	pm3.27bda 421	. . . . . 6 |- (((T e. Cat /\ U e. Cat /\ F e. (Func` <.T, U>.)) /\ F e. (Func` <.T, U>.)) -> (A.a e. dom (id` T)E.b e. dom (id` U)(F` (I1` a)) = (I2` b) /\ (A.m e. M1 (F` (I1` ((dom` T)` m))) = (I2` ((dom` U)` (F` m))) /\ A.m e. M1 (F` (I1` (C1` m))) = (I2` (C2` (F` m)))) /\ A.m e. M1 A.n e. M1 ((C1` n) = ((dom` T)` m) -> (F` (m(o` T)n)) = ((F` m)(o` U)(F` n)))))
15	14	3simp2d 795	. . . . 5 |- (((T e. Cat /\ U e. Cat /\ F e. (Func` <.T, U>.)) /\ F e. (Func` <.T, U>.)) -> (A.m e. M1 (F` (I1` ((dom` T)` m))) = (I2` ((dom` U)` (F` m))) /\ A.m e. M1 (F` (I1` (C1` m))) = (I2` (C2` (F` m)))))
16	15	pm3.27d 325	. . . 4 |- (((T e. Cat /\ U e. Cat /\ F e. (Func` <.T, U>.)) /\ F e. (Func` <.T, U>.)) -> A.m e. M1 (F` (I1` (C1` m))) = (I2` (C2` (F` m))))
17	16	ex 373	. . 3 |- ((T e. Cat /\ U e. Cat /\ F e. (Func` <.T, U>.)) -> (F e. (Func` <.T, U>.) -> A.m e. M1 (F` (I1` (C1` m))) = (I2` (C2` (F` m)))))
18	17	3expia 835	. 2 |- ((T e. Cat /\ U e. Cat) -> (F e. (Func` <.T, U>.) -> (F e. (Func` <.T, U>.) -> A.m e. M1 (F` (I1` (C1` m))) = (I2` (C2` (F` m))))))
19	18	pm2.43d 65	1 |- ((T e. Cat /\ U e. Cat) -> (F e. (Func` <.T, U>.) -> A.m e. M1 (F` (I1` (C1` m))) = (I2` (C2` (F` m)))))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646  <.cop 2411  dom cdm 3170  -->wf 3178  ` cfv 3182  (class class class)co 3963  domcdom_ 10644  codccod_ 10645  idcid_ 10646  oco_ 10647  Catccat 10685  Funccfunc 10751

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-oprab 3966  df-map 4324  df-func 10753

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