Theorem enmap1lem5 6074

Description: Lemma for enmap2 6069. Calculate the range of W. (Contributed by SF, 3-Mar-2015.)

Hypothesis

Ref	Expression
enmap1lem5.1	|- W = (s e. (A ^m G) |-> (r o. s))

Assertion

Ref	Expression
enmap1lem5	|- (r:A-1-1-onto->B -> ran W = (B ^m G))

Distinct variable groups:   G,s   s,r   A,s

Allowed substitution hints:   A(r)   B(s,r)   G(r)   W(s,r)

Proof of Theorem enmap1lem5

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		enmap1lem5.1	. . . 4 |- W = (s e. (A ^m G) |-> (r o. s))
2	1	enmap1lem2 6071	. . 3 |- W Fn (A ^m G)
3		coeq2 4876	. . . . . . 7 |- (s = p -> (r o. s) = (r o. p))
4		vex 2863	. . . . . . . 8 |- r e. _V
5		vex 2863	. . . . . . . 8 |- p e. _V
6	4, 5	coex 4751	. . . . . . 7 |- (r o. p) e. _V
7	3, 1, 6	fvmpt 5701	. . . . . 6 |- (p e. (A ^m G) -> (W` p) = (r o. p))
8	7	adantl 452	. . . . 5 |- ((r:A-1-1-onto->B /\ p e. (A ^m G)) -> (W` p) = (r o. p))
9		f1of 5288	. . . . . . 7 |- (r:A-1-1-onto->B -> r:A-->B)
10		elmapi 6017	. . . . . . 7 |- (p e. (A ^m G) -> p:G-->A)
11		fco 5232	. . . . . . 7 |- ((r:A-->B /\ p:G-->A) -> (r o. p):G-->B)
12	9, 10, 11	syl2an 463	. . . . . 6 |- ((r:A-1-1-onto->B /\ p e. (A ^m G)) -> (r o. p):G-->B)
13		f1ofo 5294	. . . . . . . . 9 |- (r:A-1-1-onto->B -> r:A-onto->B)
14		forn 5273	. . . . . . . . 9 |- (r:A-onto->B -> ran r = B)
15	13, 14	syl 15	. . . . . . . 8 |- (r:A-1-1-onto->B -> ran r = B)
16	4	rnex 5108	. . . . . . . 8 |- ran r e. _V
17	15, 16	syl6eqelr 2442	. . . . . . 7 |- (r:A-1-1-onto->B -> B e. _V)
18		elovex2 5651	. . . . . . 7 |- (p e. (A ^m G) -> G e. _V)
19		elmapg 6013	. . . . . . . 8 |- ((B e. _V /\ G e. _V /\ (r o. p) e. _V) -> ((r o. p) e. (B ^m G) <-> (r o. p):G-->B))
20	6, 19	mp3an3 1266	. . . . . . 7 |- ((B e. _V /\ G e. _V) -> ((r o. p) e. (B ^m G) <-> (r o. p):G-->B))
21	17, 18, 20	syl2an 463	. . . . . 6 |- ((r:A-1-1-onto->B /\ p e. (A ^m G)) -> ((r o. p) e. (B ^m G) <-> (r o. p):G-->B))
22	12, 21	mpbird 223	. . . . 5 |- ((r:A-1-1-onto->B /\ p e. (A ^m G)) -> (r o. p) e. (B ^m G))
23	8, 22	eqeltrd 2427	. . . 4 |- ((r:A-1-1-onto->B /\ p e. (A ^m G)) -> (W` p) e. (B ^m G))
24	23	ralrimiva 2698	. . 3 |- (r:A-1-1-onto->B -> A.p e. (A ^m G)(W` p) e. (B ^m G))
25		fnfvrnss 5430	. . 3 |- ((W Fn (A ^m G) /\ A.p e. (A ^m G)(W` p) e. (B ^m G)) -> ran W C_ (B ^m G))
26	2, 24, 25	sylancr 644	. 2 |- (r:A-1-1-onto->B -> ran W C_ (B ^m G))
27		f1ocnv 5300	. . . . . . . . . . 11 |- (r:A-1-1-onto->B -> `'r:B-1-1-onto->A)
28		f1of 5288	. . . . . . . . . . 11 |- (`'r:B-1-1-onto->A -> `'r:B-->A)
29	27, 28	syl 15	. . . . . . . . . 10 |- (r:A-1-1-onto->B -> `'r:B-->A)
30		elmapi 6017	. . . . . . . . . 10 |- (p e. (B ^m G) -> p:G-->B)
31		fco 5232	. . . . . . . . . 10 |- ((`'r:B-->A /\ p:G-->B) -> (`'r o. p):G-->A)
32	29, 30, 31	syl2an 463	. . . . . . . . 9 |- ((r:A-1-1-onto->B /\ p e. (B ^m G)) -> (`'r o. p):G-->A)
33		f1odm 5291	. . . . . . . . . . 11 |- (r:A-1-1-onto->B -> dom r = A)
34	4	dmex 5107	. . . . . . . . . . 11 |- dom r e. _V
35	33, 34	syl6eqelr 2442	. . . . . . . . . 10 |- (r:A-1-1-onto->B -> A e. _V)
36		elovex2 5651	. . . . . . . . . 10 |- (p e. (B ^m G) -> G e. _V)
37	4	cnvex 5103	. . . . . . . . . . . 12 |- `'r e. _V
38	37, 5	coex 4751	. . . . . . . . . . 11 |- (`'r o. p) e. _V
39		elmapg 6013	. . . . . . . . . . 11 |- ((A e. _V /\ G e. _V /\ (`'r o. p) e. _V) -> ((`'r o. p) e. (A ^m G) <-> (`'r o. p):G-->A))
40	38, 39	mp3an3 1266	. . . . . . . . . 10 |- ((A e. _V /\ G e. _V) -> ((`'r o. p) e. (A ^m G) <-> (`'r o. p):G-->A))
41	35, 36, 40	syl2an 463	. . . . . . . . 9 |- ((r:A-1-1-onto->B /\ p e. (B ^m G)) -> ((`'r o. p) e. (A ^m G) <-> (`'r o. p):G-->A))
42	32, 41	mpbird 223	. . . . . . . 8 |- ((r:A-1-1-onto->B /\ p e. (B ^m G)) -> (`'r o. p) e. (A ^m G))
43		coeq2 4876	. . . . . . . . 9 |- (s = (`'r o. p) -> (r o. s) = (r o. (`'r o. p)))
44	4, 38	coex 4751	. . . . . . . . 9 |- (r o. (`'r o. p)) e. _V
45	43, 1, 44	fvmpt 5701	. . . . . . . 8 |- ((`'r o. p) e. (A ^m G) -> (W` (`'r o. p)) = (r o. (`'r o. p)))
46	42, 45	syl 15	. . . . . . 7 |- ((r:A-1-1-onto->B /\ p e. (B ^m G)) -> (W` (`'r o. p)) = (r o. (`'r o. p)))
47		coass 5098	. . . . . . . 8 |- ((r o. `'r) o. p) = (r o. (`'r o. p))
48		f1ococnv2 5310	. . . . . . . . . 10 |- (r:A-1-1-onto->B -> (r o. `'r) = ( _I |` B))
49	48	coeq1d 4879	. . . . . . . . 9 |- (r:A-1-1-onto->B -> ((r o. `'r) o. p) = (( _I |` B) o. p))
50		fcoi2 5242	. . . . . . . . . 10 |- (p:G-->B -> (( _I |` B) o. p) = p)
51	30, 50	syl 15	. . . . . . . . 9 |- (p e. (B ^m G) -> (( _I |` B) o. p) = p)
52	49, 51	sylan9eq 2405	. . . . . . . 8 |- ((r:A-1-1-onto->B /\ p e. (B ^m G)) -> ((r o. `'r) o. p) = p)
53	47, 52	syl5eqr 2399	. . . . . . 7 |- ((r:A-1-1-onto->B /\ p e. (B ^m G)) -> (r o. (`'r o. p)) = p)
54	46, 53	eqtrd 2385	. . . . . 6 |- ((r:A-1-1-onto->B /\ p e. (B ^m G)) -> (W` (`'r o. p)) = p)
55		fnbrfvb 5359	. . . . . . 7 |- ((W Fn (A ^m G) /\ (`'r o. p) e. (A ^m G)) -> ((W` (`'r o. p)) = p <-> (`'r o. p)Wp))
56	2, 42, 55	sylancr 644	. . . . . 6 |- ((r:A-1-1-onto->B /\ p e. (B ^m G)) -> ((W` (`'r o. p)) = p <-> (`'r o. p)Wp))
57	54, 56	mpbid 201	. . . . 5 |- ((r:A-1-1-onto->B /\ p e. (B ^m G)) -> (`'r o. p)Wp)
58		brelrn 4961	. . . . 5 |- ((`'r o. p)Wp -> p e. ran W)
59	57, 58	syl 15	. . . 4 |- ((r:A-1-1-onto->B /\ p e. (B ^m G)) -> p e. ran W)
60	59	ex 423	. . 3 |- (r:A-1-1-onto->B -> (p e. (B ^m G) -> p e. ran W))
61	60	ssrdv 3279	. 2 |- (r:A-1-1-onto->B -> (B ^m G) C_ ran W)
62	26, 61	eqssd 3290	1 |- (r:A-1-1-onto->B -> ran W = (B ^m G))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  A.wral 2615  _Vcvv 2860   C_ wss 3258   class class class wbr 4640   o. ccom 4722   _I cid 4764  `'ccnv 4772  dom cdm 4773  ran crn 4774   |` cres 4775   Fn wfn 4777  -->wf 4778  -onto->wfo 4780  -1-1-onto->wf1o 4781  ` cfv 4782  (class class class)co 5526   |-> cmpt 5652   ^m cmap 6000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-map 6002

This theorem is referenced by:  enmap1  6075