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Theorem fmp 15188
Description: Functors preserve morphisms composition. JFM CAT1 th. 99.
Hypotheses
Ref Expression
fmp.1 |- M1 = dom (dom` T)
fmp.2 |- C1 = (cod` T)
fmp.3 |- D1 = (dom` T)
fmp.6 |- Ro1 = (o` T)
fmp.7 |- Ro2 = (o` U)
Assertion
Ref Expression
fmp |- ((T e. Cat /\ U e. Cat ) -> (F e. ( Func ` <.T, U>.) -> A.m e. M1 A.n e. M1 ((C1` n) = (D1` m) -> (F` (mRo1n)) = ((F` m)Ro2(F` n)))))
Distinct variable groups:   m,F,n   m,M1   T,m,n   U,m,n
Proof of Theorem fmp
StepHypRef Expression
1 eqid 1884 . . . . . . . . 9 |- dom (id` T) = dom (id` T)
2 fmp.1 . . . . . . . . 9 |- M1 = dom (dom` T)
3 fmp.3 . . . . . . . . 9 |- D1 = (dom` T)
4 fmp.2 . . . . . . . . 9 |- C1 = (cod` T)
5 eqid 1884 . . . . . . . . 9 |- (id` T) = (id` T)
6 fmp.6 . . . . . . . . 9 |- Ro1 = (o` T)
7 eqid 1884 . . . . . . . . 9 |- dom (id` U) = dom (id` U)
8 eqid 1884 . . . . . . . . 9 |- dom (dom` U) = dom (dom` U)
9 eqid 1884 . . . . . . . . 9 |- (dom` U) = (dom` U)
10 eqid 1884 . . . . . . . . 9 |- (cod` U) = (cod` U)
11 eqid 1884 . . . . . . . . 9 |- (id` U) = (id` U)
12 fmp.7 . . . . . . . . 9 |- Ro2 = (o` U)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12isfunb 15183 . . . . . . . 8 |- ((T e. Cat /\ U e. Cat /\ F e. ( Func ` <.T, U>.)) -> (F e. ( Func ` <.T, U>.) <-> (F:M1-->dom (dom` U) /\ (A.x e. dom (id` T)E.y e. dom (id` U)(F` ((id` T)` x)) = ((id` U)` y) /\ (A.m e. M1 (F` ((id` T)` (D1` m))) = ((id` U)` ((dom` U)` (F` m))) /\ A.m e. M1 (F` ((id` T)` (C1` m))) = ((id` U)` ((cod` U)` (F` m)))) /\ A.m e. M1 A.n e. M1 ((C1` n) = (D1` m) -> (F` (mRo1n)) = ((F` m)Ro2(F` n)))))))
1413biimpd 170 . . . . . . 7 |- ((T e. Cat /\ U e. Cat /\ F e. ( Func ` <.T, U>.)) -> (F e. ( Func ` <.T, U>.) -> (F:M1-->dom (dom` U) /\ (A.x e. dom (id` T)E.y e. dom (id` U)(F` ((id` T)` x)) = ((id` U)` y) /\ (A.m e. M1 (F` ((id` T)` (D1` m))) = ((id` U)` ((dom` U)` (F` m))) /\ A.m e. M1 (F` ((id` T)` (C1` m))) = ((id` U)` ((cod` U)` (F` m)))) /\ A.m e. M1 A.n e. M1 ((C1` n) = (D1` m) -> (F` (mRo1n)) = ((F` m)Ro2(F` n)))))))
15143expia 1069 . . . . . 6 |- ((T e. Cat /\ U e. Cat ) -> (F e. ( Func ` <.T, U>.) -> (F e. ( Func ` <.T, U>.) -> (F:M1-->dom (dom` U) /\ (A.x e. dom (id` T)E.y e. dom (id` U)(F` ((id` T)` x)) = ((id` U)` y) /\ (A.m e. M1 (F` ((id` T)` (D1` m))) = ((id` U)` ((dom` U)` (F` m))) /\ A.m e. M1 (F` ((id` T)` (C1` m))) = ((id` U)` ((cod` U)` (F` m)))) /\ A.m e. M1 A.n e. M1 ((C1` n) = (D1` m) -> (F` (mRo1n)) = ((F` m)Ro2(F` n))))))))
1615pm2.43d 79 . . . . 5 |- ((T e. Cat /\ U e. Cat ) -> (F e. ( Func ` <.T, U>.) -> (F:M1-->dom (dom` U) /\ (A.x e. dom (id` T)E.y e. dom (id` U)(F` ((id` T)` x)) = ((id` U)` y) /\ (A.m e. M1 (F` ((id` T)` (D1` m))) = ((id` U)` ((dom` U)` (F` m))) /\ A.m e. M1 (F` ((id` T)` (C1` m))) = ((id` U)` ((cod` U)` (F` m)))) /\ A.m e. M1 A.n e. M1 ((C1` n) = (D1` m) -> (F` (mRo1n)) = ((F` m)Ro2(F` n)))))))
1716imp 377 . . . 4 |- (((T e. Cat /\ U e. Cat ) /\ F e. ( Func ` <.T, U>.)) -> (F:M1-->dom (dom` U) /\ (A.x e. dom (id` T)E.y e. dom (id` U)(F` ((id` T)` x)) = ((id` U)` y) /\ (A.m e. M1 (F` ((id` T)` (D1` m))) = ((id` U)` ((dom` U)` (F` m))) /\ A.m e. M1 (F` ((id` T)` (C1` m))) = ((id` U)` ((cod` U)` (F` m)))) /\ A.m e. M1 A.n e. M1 ((C1` n) = (D1` m) -> (F` (mRo1n)) = ((F` m)Ro2(F` n))))))
1817simprd 352 . . 3 |- (((T e. Cat /\ U e. Cat ) /\ F e. ( Func ` <.T, U>.)) -> (A.x e. dom (id` T)E.y e. dom (id` U)(F` ((id` T)` x)) = ((id` U)` y) /\ (A.m e. M1 (F` ((id` T)` (D1` m))) = ((id` U)` ((dom` U)` (F` m))) /\ A.m e. M1 (F` ((id` T)` (C1` m))) = ((id` U)` ((cod` U)` (F` m)))) /\ A.m e. M1 A.n e. M1 ((C1` n) = (D1` m) -> (F` (mRo1n)) = ((F` m)Ro2(F` n)))))
1918simp3d 890 . 2 |- (((T e. Cat /\ U e. Cat ) /\ F e. ( Func ` <.T, U>.)) -> A.m e. M1 A.n e. M1 ((C1` n) = (D1` m) -> (F` (mRo1n)) = ((F` m)Ro2(F` n))))
2019ex 402 1 |- ((T e. Cat /\ U e. Cat ) -> (F e. ( Func ` <.T, U>.) -> A.m e. M1 A.n e. M1 ((C1` n) = (D1` m) -> (F` (mRo1n)) = ((F` m)Ro2(F` n)))))
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  <.cop 3046  dom cdm 3986  -->wf 3994  ` cfv 3998  (class class class)co 4884  domcdom_ 15059  codccod_ 15060  idcid_ 15061  oco_ 15062   Cat ccat 15099   Func cfunc 15179
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790
This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-map 5383  df-func 15181
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