Theorem enmap2lem5 6067

Description: Lemma for enmap2 6068. Calculate the range of W. (Contributed by SF, 26-Feb-2015.)

Hypothesis

Ref	Expression
enmap2lem5.1	⊢ W = (s ∈ (G ↑m a) ↦ (s ∘ ◡r))

Assertion

Ref	Expression
enmap2lem5	⊢ (r:a–1-1-onto→b → ran W = (G ↑m b))

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

Allowed substitution hints:   G(r,a,b)   W(s,r,a,b)

Proof of Theorem enmap2lem5

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		enmap2lem5.1	. . . 4 ⊢ W = (s ∈ (G ↑m a) ↦ (s ∘ ◡r))
2	1	enmap2lem2 6064	. . 3 ⊢ W Fn (G ↑m a)
3		coeq1 4874	. . . . . . 7 ⊢ (s = p → (s ∘ ◡r) = (p ∘ ◡r))
4		vex 2862	. . . . . . . 8 ⊢ p ∈ V
5		vex 2862	. . . . . . . . 9 ⊢ r ∈ V
6	5	cnvex 5102	. . . . . . . 8 ⊢ ◡r ∈ V
7	4, 6	coex 4750	. . . . . . 7 ⊢ (p ∘ ◡r) ∈ V
8	3, 1, 7	fvmpt 5700	. . . . . 6 ⊢ (p ∈ (G ↑m a) → (W ‘p) = (p ∘ ◡r))
9	8	adantl 452	. . . . 5 ⊢ ((r:a–1-1-onto→b ∧ p ∈ (G ↑m a)) → (W ‘p) = (p ∘ ◡r))
10		elmapi 6016	. . . . . . 7 ⊢ (p ∈ (G ↑m a) → p:a–→G)
11		f1ocnv 5299	. . . . . . . 8 ⊢ (r:a–1-1-onto→b → ◡r:b–1-1-onto→a)
12		f1of 5287	. . . . . . . 8 ⊢ (◡r:b–1-1-onto→a → ◡r:b–→a)
13	11, 12	syl 15	. . . . . . 7 ⊢ (r:a–1-1-onto→b → ◡r:b–→a)
14		fco 5231	. . . . . . 7 ⊢ ((p:a–→G ∧ ◡r:b–→a) → (p ∘ ◡r):b–→G)
15	10, 13, 14	syl2anr 464	. . . . . 6 ⊢ ((r:a–1-1-onto→b ∧ p ∈ (G ↑m a)) → (p ∘ ◡r):b–→G)
16		elovex1 5649	. . . . . . . 8 ⊢ (p ∈ (G ↑m a) → G ∈ V)
17		vex 2862	. . . . . . . . 9 ⊢ b ∈ V
18		elmapg 6012	. . . . . . . . 9 ⊢ ((G ∈ V ∧ b ∈ V ∧ (p ∘ ◡r) ∈ V) → ((p ∘ ◡r) ∈ (G ↑m b) ↔ (p ∘ ◡r):b–→G))
19	17, 7, 18	mp3an23 1269	. . . . . . . 8 ⊢ (G ∈ V → ((p ∘ ◡r) ∈ (G ↑m b) ↔ (p ∘ ◡r):b–→G))
20	16, 19	syl 15	. . . . . . 7 ⊢ (p ∈ (G ↑m a) → ((p ∘ ◡r) ∈ (G ↑m b) ↔ (p ∘ ◡r):b–→G))
21	20	adantl 452	. . . . . 6 ⊢ ((r:a–1-1-onto→b ∧ p ∈ (G ↑m a)) → ((p ∘ ◡r) ∈ (G ↑m b) ↔ (p ∘ ◡r):b–→G))
22	15, 21	mpbird 223	. . . . 5 ⊢ ((r:a–1-1-onto→b ∧ p ∈ (G ↑m a)) → (p ∘ ◡r) ∈ (G ↑m b))
23	9, 22	eqeltrd 2427	. . . 4 ⊢ ((r:a–1-1-onto→b ∧ p ∈ (G ↑m a)) → (W ‘p) ∈ (G ↑m b))
24	23	ralrimiva 2697	. . 3 ⊢ (r:a–1-1-onto→b → ∀p ∈ (G ↑m a)(W ‘p) ∈ (G ↑m b))
25		fnfvrnss 5429	. . 3 ⊢ ((W Fn (G ↑m a) ∧ ∀p ∈ (G ↑m a)(W ‘p) ∈ (G ↑m b)) → ran W ⊆ (G ↑m b))
26	2, 24, 25	sylancr 644	. 2 ⊢ (r:a–1-1-onto→b → ran W ⊆ (G ↑m b))
27		elmapi 6016	. . . . . . . . . 10 ⊢ (p ∈ (G ↑m b) → p:b–→G)
28		f1of 5287	. . . . . . . . . 10 ⊢ (r:a–1-1-onto→b → r:a–→b)
29		fco 5231	. . . . . . . . . 10 ⊢ ((p:b–→G ∧ r:a–→b) → (p ∘ r):a–→G)
30	27, 28, 29	syl2anr 464	. . . . . . . . 9 ⊢ ((r:a–1-1-onto→b ∧ p ∈ (G ↑m b)) → (p ∘ r):a–→G)
31		elovex1 5649	. . . . . . . . . . 11 ⊢ (p ∈ (G ↑m b) → G ∈ V)
32		vex 2862	. . . . . . . . . . . 12 ⊢ a ∈ V
33	4, 5	coex 4750	. . . . . . . . . . . 12 ⊢ (p ∘ r) ∈ V
34		elmapg 6012	. . . . . . . . . . . 12 ⊢ ((G ∈ V ∧ a ∈ V ∧ (p ∘ r) ∈ V) → ((p ∘ r) ∈ (G ↑m a) ↔ (p ∘ r):a–→G))
35	32, 33, 34	mp3an23 1269	. . . . . . . . . . 11 ⊢ (G ∈ V → ((p ∘ r) ∈ (G ↑m a) ↔ (p ∘ r):a–→G))
36	31, 35	syl 15	. . . . . . . . . 10 ⊢ (p ∈ (G ↑m b) → ((p ∘ r) ∈ (G ↑m a) ↔ (p ∘ r):a–→G))
37	36	adantl 452	. . . . . . . . 9 ⊢ ((r:a–1-1-onto→b ∧ p ∈ (G ↑m b)) → ((p ∘ r) ∈ (G ↑m a) ↔ (p ∘ r):a–→G))
38	30, 37	mpbird 223	. . . . . . . 8 ⊢ ((r:a–1-1-onto→b ∧ p ∈ (G ↑m b)) → (p ∘ r) ∈ (G ↑m a))
39		coeq1 4874	. . . . . . . . . 10 ⊢ (s = (p ∘ r) → (s ∘ ◡r) = ((p ∘ r) ∘ ◡r))
40		coass 5097	. . . . . . . . . 10 ⊢ ((p ∘ r) ∘ ◡r) = (p ∘ (r ∘ ◡r))
41	39, 40	syl6eq 2401	. . . . . . . . 9 ⊢ (s = (p ∘ r) → (s ∘ ◡r) = (p ∘ (r ∘ ◡r)))
42	5, 6	coex 4750	. . . . . . . . . 10 ⊢ (r ∘ ◡r) ∈ V
43	4, 42	coex 4750	. . . . . . . . 9 ⊢ (p ∘ (r ∘ ◡r)) ∈ V
44	41, 1, 43	fvmpt 5700	. . . . . . . 8 ⊢ ((p ∘ r) ∈ (G ↑m a) → (W ‘(p ∘ r)) = (p ∘ (r ∘ ◡r)))
45	38, 44	syl 15	. . . . . . 7 ⊢ ((r:a–1-1-onto→b ∧ p ∈ (G ↑m b)) → (W ‘(p ∘ r)) = (p ∘ (r ∘ ◡r)))
46		f1ococnv2 5309	. . . . . . . . 9 ⊢ (r:a–1-1-onto→b → (r ∘ ◡r) = ( I ↾ b))
47	46	coeq2d 4879	. . . . . . . 8 ⊢ (r:a–1-1-onto→b → (p ∘ (r ∘ ◡r)) = (p ∘ ( I ↾ b)))
48		fcoi1 5240	. . . . . . . . 9 ⊢ (p:b–→G → (p ∘ ( I ↾ b)) = p)
49	27, 48	syl 15	. . . . . . . 8 ⊢ (p ∈ (G ↑m b) → (p ∘ ( I ↾ b)) = p)
50	47, 49	sylan9eq 2405	. . . . . . 7 ⊢ ((r:a–1-1-onto→b ∧ p ∈ (G ↑m b)) → (p ∘ (r ∘ ◡r)) = p)
51	45, 50	eqtrd 2385	. . . . . 6 ⊢ ((r:a–1-1-onto→b ∧ p ∈ (G ↑m b)) → (W ‘(p ∘ r)) = p)
52		fnbrfvb 5358	. . . . . . 7 ⊢ ((W Fn (G ↑m a) ∧ (p ∘ r) ∈ (G ↑m a)) → ((W ‘(p ∘ r)) = p ↔ (p ∘ r)Wp))
53	2, 38, 52	sylancr 644	. . . . . 6 ⊢ ((r:a–1-1-onto→b ∧ p ∈ (G ↑m b)) → ((W ‘(p ∘ r)) = p ↔ (p ∘ r)Wp))
54	51, 53	mpbid 201	. . . . 5 ⊢ ((r:a–1-1-onto→b ∧ p ∈ (G ↑m b)) → (p ∘ r)Wp)
55		brelrn 4960	. . . . 5 ⊢ ((p ∘ r)Wp → p ∈ ran W)
56	54, 55	syl 15	. . . 4 ⊢ ((r:a–1-1-onto→b ∧ p ∈ (G ↑m b)) → p ∈ ran W)
57	56	ex 423	. . 3 ⊢ (r:a–1-1-onto→b → (p ∈ (G ↑m b) → p ∈ ran W))
58	57	ssrdv 3278	. 2 ⊢ (r:a–1-1-onto→b → (G ↑m b) ⊆ ran W)
59	26, 58	eqssd 3289	1 ⊢ (r:a–1-1-onto→b → ran W = (G ↑m b))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  Vcvv 2859   ⊆ wss 3257   class class class wbr 4639   ∘ ccom 4721   I cid 4763  ^◡ccnv 4771  ran crn 4773   ↾ cres 4774   Fn wfn 4776  –→wf 4777  –1-1-onto→wf1o 4780   ‘cfv 4781  (class class class)co 5525   ↦ cmpt 5651   ↑_m cmap 5999

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-map 6001

This theorem is referenced by:  enmap2  6068