Theorem enmap2lem3 6066

Description: Lemma for enmap2 6069. Binary relationship condition over W. (Contributed by SF, 26-Feb-2015.)

Hypothesis

Ref	Expression
enmap2lem3.1	⊢ W = (s ∈ (G ↑m a) ↦ (s ∘ ◡r))

Assertion

Ref	Expression
enmap2lem3	⊢ (r:a–1-1-onto→b → (SWT → S = (T ∘ r)))

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

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

Proof of Theorem enmap2lem3

Step	Hyp	Ref	Expression
1		breldm 4912	. . . 4 ⊢ (SWT → S ∈ dom W)
2		enmap2lem3.1	. . . . . 6 ⊢ W = (s ∈ (G ↑m a) ↦ (s ∘ ◡r))
3	2	enmap2lem2 6065	. . . . 5 ⊢ W Fn (G ↑m a)
4		fndm 5183	. . . . 5 ⊢ (W Fn (G ↑m a) → dom W = (G ↑m a))
5	3, 4	ax-mp 5	. . . 4 ⊢ dom W = (G ↑m a)
6	1, 5	syl6eleq 2443	. . 3 ⊢ (SWT → S ∈ (G ↑m a))
7		fnbrfvb 5359	. . . . . 6 ⊢ ((W Fn (G ↑m a) ∧ S ∈ (G ↑m a)) → ((W ‘S) = T ↔ SWT))
8	3, 7	mpan 651	. . . . 5 ⊢ (S ∈ (G ↑m a) → ((W ‘S) = T ↔ SWT))
9		vex 2863	. . . . . . . . . . 11 ⊢ r ∈ V
10	9	cnvex 5103	. . . . . . . . . 10 ⊢ ◡r ∈ V
11		coexg 4750	. . . . . . . . . 10 ⊢ ((S ∈ (G ↑m a) ∧ ◡r ∈ V) → (S ∘ ◡r) ∈ V)
12	10, 11	mpan2 652	. . . . . . . . 9 ⊢ (S ∈ (G ↑m a) → (S ∘ ◡r) ∈ V)
13		coeq1 4875	. . . . . . . . . 10 ⊢ (s = S → (s ∘ ◡r) = (S ∘ ◡r))
14	13, 2	fvmptg 5699	. . . . . . . . 9 ⊢ ((S ∈ (G ↑m a) ∧ (S ∘ ◡r) ∈ V) → (W ‘S) = (S ∘ ◡r))
15	12, 14	mpdan 649	. . . . . . . 8 ⊢ (S ∈ (G ↑m a) → (W ‘S) = (S ∘ ◡r))
16	15	eqeq1d 2361	. . . . . . 7 ⊢ (S ∈ (G ↑m a) → ((W ‘S) = T ↔ (S ∘ ◡r) = T))
17		eqcom 2355	. . . . . . 7 ⊢ ((S ∘ ◡r) = T ↔ T = (S ∘ ◡r))
18	16, 17	syl6bb 252	. . . . . 6 ⊢ (S ∈ (G ↑m a) → ((W ‘S) = T ↔ T = (S ∘ ◡r)))
19	18	biimpd 198	. . . . 5 ⊢ (S ∈ (G ↑m a) → ((W ‘S) = T → T = (S ∘ ◡r)))
20	8, 19	sylbird 226	. . . 4 ⊢ (S ∈ (G ↑m a) → (SWT → T = (S ∘ ◡r)))
21	6, 20	mpcom 32	. . 3 ⊢ (SWT → T = (S ∘ ◡r))
22	6, 21	jca 518	. 2 ⊢ (SWT → (S ∈ (G ↑m a) ∧ T = (S ∘ ◡r)))
23		f1ococnv1 5311	. . . . . . 7 ⊢ (r:a–1-1-onto→b → (◡r ∘ r) = ( I ↾ a))
24	23	coeq2d 4880	. . . . . 6 ⊢ (r:a–1-1-onto→b → (S ∘ (◡r ∘ r)) = (S ∘ ( I ↾ a)))
25	24	adantr 451	. . . . 5 ⊢ ((r:a–1-1-onto→b ∧ S ∈ (G ↑m a)) → (S ∘ (◡r ∘ r)) = (S ∘ ( I ↾ a)))
26		elmapi 6017	. . . . . . 7 ⊢ (S ∈ (G ↑m a) → S:a–→G)
27		fcoi1 5241	. . . . . . 7 ⊢ (S:a–→G → (S ∘ ( I ↾ a)) = S)
28	26, 27	syl 15	. . . . . 6 ⊢ (S ∈ (G ↑m a) → (S ∘ ( I ↾ a)) = S)
29	28	adantl 452	. . . . 5 ⊢ ((r:a–1-1-onto→b ∧ S ∈ (G ↑m a)) → (S ∘ ( I ↾ a)) = S)
30	25, 29	eqtr2d 2386	. . . 4 ⊢ ((r:a–1-1-onto→b ∧ S ∈ (G ↑m a)) → S = (S ∘ (◡r ∘ r)))
31		coeq1 4875	. . . . . 6 ⊢ (T = (S ∘ ◡r) → (T ∘ r) = ((S ∘ ◡r) ∘ r))
32		coass 5098	. . . . . 6 ⊢ ((S ∘ ◡r) ∘ r) = (S ∘ (◡r ∘ r))
33	31, 32	syl6eq 2401	. . . . 5 ⊢ (T = (S ∘ ◡r) → (T ∘ r) = (S ∘ (◡r ∘ r)))
34	33	eqeq2d 2364	. . . 4 ⊢ (T = (S ∘ ◡r) → (S = (T ∘ r) ↔ S = (S ∘ (◡r ∘ r))))
35	30, 34	syl5ibrcom 213	. . 3 ⊢ ((r:a–1-1-onto→b ∧ S ∈ (G ↑m a)) → (T = (S ∘ ◡r) → S = (T ∘ r)))
36	35	expimpd 586	. 2 ⊢ (r:a–1-1-onto→b → ((S ∈ (G ↑m a) ∧ T = (S ∘ ◡r)) → S = (T ∘ r)))
37	22, 36	syl5 28	1 ⊢ (r:a–1-1-onto→b → (SWT → S = (T ∘ r)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2860   class class class wbr 4640   ∘ ccom 4722   I cid 4764  ^◡ccnv 4772  dom cdm 4773   ↾ cres 4775   Fn wfn 4777  –→wf 4778  –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:  enmap2lem4  6067