Theorem enmap1lem3 6072

Description: Lemma for enmap2 6069. Binary relationship condition over W. (Contributed by SF, 3-Mar-2015.)

Hypothesis

Ref	Expression
enmap1lem3.1	⊢ W = (s ∈ (A ↑m G) ↦ (r ∘ s))

Assertion

Ref	Expression
enmap1lem3	⊢ (r:A–1-1-onto→B → (SWT → S = (◡r ∘ T)))

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

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

Proof of Theorem enmap1lem3

Step	Hyp	Ref	Expression
1		breldm 4912	. . . 4 ⊢ (SWT → S ∈ dom W)
2		enmap1lem3.1	. . . . . 6 ⊢ W = (s ∈ (A ↑m G) ↦ (r ∘ s))
3	2	enmap1lem2 6071	. . . . 5 ⊢ W Fn (A ↑m G)
4		fndm 5183	. . . . 5 ⊢ (W Fn (A ↑m G) → dom W = (A ↑m G))
5	3, 4	ax-mp 5	. . . 4 ⊢ dom W = (A ↑m G)
6	1, 5	syl6eleq 2443	. . 3 ⊢ (SWT → S ∈ (A ↑m G))
7		fnbrfvb 5359	. . . . . . 7 ⊢ ((W Fn (A ↑m G) ∧ S ∈ (A ↑m G)) → ((W ‘S) = T ↔ SWT))
8	3, 7	mpan 651	. . . . . 6 ⊢ (S ∈ (A ↑m G) → ((W ‘S) = T ↔ SWT))
9		vex 2863	. . . . . . . . 9 ⊢ r ∈ V
10		coexg 4750	. . . . . . . . 9 ⊢ ((r ∈ V ∧ S ∈ (A ↑m G)) → (r ∘ S) ∈ V)
11	9, 10	mpan 651	. . . . . . . 8 ⊢ (S ∈ (A ↑m G) → (r ∘ S) ∈ V)
12		coeq2 4876	. . . . . . . . 9 ⊢ (s = S → (r ∘ s) = (r ∘ S))
13	12, 2	fvmptg 5699	. . . . . . . 8 ⊢ ((S ∈ (A ↑m G) ∧ (r ∘ S) ∈ V) → (W ‘S) = (r ∘ S))
14	11, 13	mpdan 649	. . . . . . 7 ⊢ (S ∈ (A ↑m G) → (W ‘S) = (r ∘ S))
15	14	eqeq1d 2361	. . . . . 6 ⊢ (S ∈ (A ↑m G) → ((W ‘S) = T ↔ (r ∘ S) = T))
16	8, 15	bitr3d 246	. . . . 5 ⊢ (S ∈ (A ↑m G) → (SWT ↔ (r ∘ S) = T))
17	16	biimpd 198	. . . 4 ⊢ (S ∈ (A ↑m G) → (SWT → (r ∘ S) = T))
18	6, 17	mpcom 32	. . 3 ⊢ (SWT → (r ∘ S) = T)
19	6, 18	jca 518	. 2 ⊢ (SWT → (S ∈ (A ↑m G) ∧ (r ∘ S) = T))
20		coass 5098	. . . . 5 ⊢ ((◡r ∘ r) ∘ S) = (◡r ∘ (r ∘ S))
21		f1ococnv1 5311	. . . . . . 7 ⊢ (r:A–1-1-onto→B → (◡r ∘ r) = ( I ↾ A))
22	21	coeq1d 4879	. . . . . 6 ⊢ (r:A–1-1-onto→B → ((◡r ∘ r) ∘ S) = (( I ↾ A) ∘ S))
23		elmapi 6017	. . . . . . 7 ⊢ (S ∈ (A ↑m G) → S:G–→A)
24		fcoi2 5242	. . . . . . 7 ⊢ (S:G–→A → (( I ↾ A) ∘ S) = S)
25	23, 24	syl 15	. . . . . 6 ⊢ (S ∈ (A ↑m G) → (( I ↾ A) ∘ S) = S)
26	22, 25	sylan9eq 2405	. . . . 5 ⊢ ((r:A–1-1-onto→B ∧ S ∈ (A ↑m G)) → ((◡r ∘ r) ∘ S) = S)
27	20, 26	syl5reqr 2400	. . . 4 ⊢ ((r:A–1-1-onto→B ∧ S ∈ (A ↑m G)) → S = (◡r ∘ (r ∘ S)))
28		coeq2 4876	. . . . 5 ⊢ ((r ∘ S) = T → (◡r ∘ (r ∘ S)) = (◡r ∘ T))
29	28	eqeq2d 2364	. . . 4 ⊢ ((r ∘ S) = T → (S = (◡r ∘ (r ∘ S)) ↔ S = (◡r ∘ T)))
30	27, 29	syl5ibcom 211	. . 3 ⊢ ((r:A–1-1-onto→B ∧ S ∈ (A ↑m G)) → ((r ∘ S) = T → S = (◡r ∘ T)))
31	30	expimpd 586	. 2 ⊢ (r:A–1-1-onto→B → ((S ∈ (A ↑m G) ∧ (r ∘ S) = T) → S = (◡r ∘ T)))
32	19, 31	syl5 28	1 ⊢ (r:A–1-1-onto→B → (SWT → S = (◡r ∘ T)))

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:  enmap1lem4  6073