Theorem tz7.48-2 3948

Description: Proposition 7.48(2) of [TakeutiZaring] p. 51.

Hypothesis

Ref	Expression
tz7.48.1	|- F Fn On

Assertion

Ref	Expression
tz7.48-2	|- (A.x e. On (F` x) e. (A \ (F"x)) -> Fun `'F)

Distinct variable groups:   x,F   x,A

Proof of Theorem tz7.48-2

Step	Hyp	Ref	Expression
1		onelon 2967	. . . . . . . . . 10 |- ((x e. On /\ y e. x) -> y e. On)
2	1	ancoms 436	. . . . . . . . 9 |- ((y e. x /\ x e. On) -> y e. On)
3		dmres 3372	. . . . . . . . . . . . . . 15 |- dom ( F |` x) = (x i^i dom F)
4	3	eleq2i 1535	. . . . . . . . . . . . . 14 |- (y e. dom ( F |` x) <-> y e. (x i^i dom F))
5		elin 2203	. . . . . . . . . . . . . 14 |- (y e. (x i^i dom F) <-> (y e. x /\ y e. dom F))
6	4, 5	bitr 173	. . . . . . . . . . . . 13 |- (y e. dom ( F |` x) <-> (y e. x /\ y e. dom F))
7		tz7.48.1	. . . . . . . . . . . . . . . 16 |- F Fn On
8		fnfun 3577	. . . . . . . . . . . . . . . 16 |- (F Fn On -> Fun F)
9	7, 8	ax-mp 7	. . . . . . . . . . . . . . 15 |- Fun F
10		funres 3543	. . . . . . . . . . . . . . 15 |- (Fun F -> Fun (F |` x))
11	9, 10	ax-mp 7	. . . . . . . . . . . . . 14 |- Fun (F |` x)
12		fvelrn 3803	. . . . . . . . . . . . . 14 |- ((Fun (F |` x) /\ y e. dom ( F |` x)) -> ((F |` x)` y) e. ran ( F |` x))
13	11, 12	mpan 694	. . . . . . . . . . . . 13 |- (y e. dom ( F |` x) -> ((F |` x)` y) e. ran ( F |` x))
14	6, 13	sylbir 201	. . . . . . . . . . . 12 |- ((y e. x /\ y e. dom F) -> ((F |` x)` y) e. ran ( F |` x))
15		fvres 3725	. . . . . . . . . . . . . . 15 |- (y e. x -> ((F |` x)` y) = (F` y))
16	15	eleq1d 1537	. . . . . . . . . . . . . 14 |- (y e. x -> (((F |` x)` y) e. ran ( F |` x) <-> (F` y) e. ran ( F |` x)))
17		df-ima 3186	. . . . . . . . . . . . . . 15 |- (F"x) = ran ( F |` x)
18	17	eleq2i 1535	. . . . . . . . . . . . . 14 |- ((F` y) e. (F"x) <-> (F` y) e. ran ( F |` x))
19	16, 18	syl6rbbr 538	. . . . . . . . . . . . 13 |- (y e. x -> ((F` y) e. (F"x) <-> ((F |` x)` y) e. ran ( F |` x)))
20	19	adantr 389	. . . . . . . . . . . 12 |- ((y e. x /\ y e. dom F) -> ((F` y) e. (F"x) <-> ((F |` x)` y) e. ran ( F |` x)))
21	14, 20	mpbird 196	. . . . . . . . . . 11 |- ((y e. x /\ y e. dom F) -> (F` y) e. (F"x))
22		eleq1a 1540	. . . . . . . . . . . 12 |- ((F` y) e. (F"x) -> ((F` x) = (F` y) -> (F` x) e. (F"x)))
23		eldifn 2159	. . . . . . . . . . . 12 |- ((F` x) e. (A \ (F"x)) -> -. (F` x) e. (F"x))
24	22, 23	nsyli 121	. . . . . . . . . . 11 |- ((F` y) e. (F"x) -> ((F` x) e. (A \ (F"x)) -> -. (F` x) = (F` y)))
25	21, 24	syl 10	. . . . . . . . . 10 |- ((y e. x /\ y e. dom F) -> ((F` x) e. (A \ (F"x)) -> -. (F` x) = (F` y)))
26		fndm 3579	. . . . . . . . . . . 12 |- (F Fn On -> dom F = On)
27	7, 26	ax-mp 7	. . . . . . . . . . 11 |- dom F = On
28	27	eleq2i 1535	. . . . . . . . . 10 |- (y e. dom F <-> y e. On)
29	25, 28	sylan2br 453	. . . . . . . . 9 |- ((y e. x /\ y e. On) -> ((F` x) e. (A \ (F"x)) -> -. (F` x) = (F` y)))
30	2, 29	syldan 467	. . . . . . . 8 |- ((y e. x /\ x e. On) -> ((F` x) e. (A \ (F"x)) -> -. (F` x) = (F` y)))
31	30	ex 373	. . . . . . 7 |- (y e. x -> (x e. On -> ((F` x) e. (A \ (F"x)) -> -. (F` x) = (F` y))))
32	31	imp3a 361	. . . . . 6 |- (y e. x -> ((x e. On /\ (F` x) e. (A \ (F"x))) -> -. (F` x) = (F` y)))
33	32	com12 11	. . . . 5 |- ((x e. On /\ (F` x) e. (A \ (F"x))) -> (y e. x -> -. (F` x) = (F` y)))
34	33	r19.21aiv 1710	. . . 4 |- ((x e. On /\ (F` x) e. (A \ (F"x))) -> A.y e. x -. (F` x) = (F` y))
35	34	r19.20ia 1702	. . 3 |- (A.x e. On (F` x) e. (A \ (F"x)) -> A.x e. On A.y e. x -. (F` x) = (F` y))
36		ssid 2076	. . . 4 |- On (_ On
37	7	tz7.48lem 3946	. . . 4 |- ((On (_ On /\ A.x e. On A.y e. x -. (F` x) = (F` y)) -> Fun `'(F |` On))
38	36, 37	mpan 694	. . 3 |- (A.x e. On A.y e. x -. (F` x) = (F` y) -> Fun `'(F |` On))
39	35, 38	syl 10	. 2 |- (A.x e. On (F` x) e. (A \ (F"x)) -> Fun `'(F |` On))
40		fnrel 3578	. . . . . 6 |- (F Fn On -> Rel F)
41	7, 40	ax-mp 7	. . . . 5 |- Rel F
42	27	eqimssi 2107	. . . . 5 |- dom F (_ On
43		relssres 3384	. . . . 5 |- ((Rel F /\ dom F (_ On) -> (F |` On) = F)
44	41, 42, 43	mp2an 696	. . . 4 |- (F |` On) = F
45		cnveq 3287	. . . 4 |- ((F |` On) = F -> `'(F |` On) = `'F)
46	44, 45	ax-mp 7	. . 3 |- `'(F |` On) = `'F
47		funeq 3527	. . 3 |- (`'(F |` On) = `'F -> (Fun `'(F |` On) <-> Fun `'F))
48	46, 47	ax-mp 7	. 2 |- (Fun `'(F |` On) <-> Fun `'F)
49	39, 48	sylib 198	1 |- (A.x e. On (F` x) e. (A \ (F"x)) -> Fun `'F)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  A.wral 1642   \ cdif 2040   i^i cin 2042   (_ wss 2043  Oncon0 2943  `'ccnv 3164  dom cdm 3165  ran crn 3166   |` cres 3167  "cima 3168  Rel wrel 3170  Fun wfun 3171   Fn wfn 3172  ` cfv 3177

This theorem is referenced by:  tz7.48-3 3949

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fv 3193

Copyright terms: Public domain