Theorem symgval 10337

Description: The value of the symmetry group function at A. (Contributed by Paul Chapman, 25-Feb-2008.)

Hypotheses

Ref	Expression
elsymgrn.1	⊢ A ∈ V
elsymgrn.2	⊢ P = {x∣x:A–1-1-onto→A}

Assertion

Ref	Expression
symgval	⊢ (SymGrp ‘A) = {⟨⟨f, g⟩, h⟩∣((f ∈ P ⋀ g ∈ P) ⋀ h = (f ∘ g))}

Distinct variable groups:   A,f,g,h,x   P,f,g,h

Proof of Theorem symgval

Step	Hyp	Ref	Expression
1		df-symgrp 10334	. . 3 ⊢ SymGrp = {⟨x, y⟩∣y = {⟨⟨f, g⟩, h⟩∣(f:x–1-1-onto→x ⋀ g:x–1-1-onto→x ⋀ h = (f ∘ g))}}
2	1	fveq1i 3716	. 2 ⊢ (SymGrp ‘A) = ({⟨x, y⟩∣y = {⟨⟨f, g⟩, h⟩∣(f:x–1-1-onto→x ⋀ g:x–1-1-onto→x ⋀ h = (f ∘ g))}} ‘A)
3		elsymgrn.1	. . 3 ⊢ A ∈ V
4		elsymgrn.2	. . . . . . 7 ⊢ P = {x∣x:A–1-1-onto→A}
5		equid 1124	. . . . . . . . 9 ⊢ x = x
6	5	biantru 723	. . . . . . . 8 ⊢ (x:A–1-1-onto→A ↔ (x:A–1-1-onto→A ⋀ x = x))
7	6	abbii 1572	. . . . . . 7 ⊢ {x∣x:A–1-1-onto→A} = {x∣(x:A–1-1-onto→A ⋀ x = x)}
8	4, 7	eqtr 1492	. . . . . 6 ⊢ P = {x∣(x:A–1-1-onto→A ⋀ x = x)}
9	8	f1oabexg 3691	. . . . 5 ⊢ ((A ∈ V ⋀ A ∈ V) → P ∈ V)
10	3, 3, 9	mp2an 696	. . . 4 ⊢ P ∈ V
11	3, 4	symgoprab 10336	. . . 4 ⊢ {⟨⟨f, g⟩, h⟩∣(f:A–1-1-onto→A ⋀ g:A–1-1-onto→A ⋀ h = (f ∘ g))} = {⟨⟨f, g⟩, h⟩∣((f ∈ P ⋀ g ∈ P) ⋀ h = (f ∘ g))}
12	10, 10, 11	oprabex2 4012	. . 3 ⊢ {⟨⟨f, g⟩, h⟩∣(f:A–1-1-onto→A ⋀ g:A–1-1-onto→A ⋀ h = (f ∘ g))} ∈ V
13		f1oeq2 3676	. . . . . 6 ⊢ (x = A → (f:x–1-1-onto→x ↔ f:A–1-1-onto→x))
14		f1oeq3 3677	. . . . . 6 ⊢ (x = A → (f:A–1-1-onto→x ↔ f:A–1-1-onto→A))
15	13, 14	bitrd 527	. . . . 5 ⊢ (x = A → (f:x–1-1-onto→x ↔ f:A–1-1-onto→A))
16		f1oeq2 3676	. . . . . 6 ⊢ (x = A → (g:x–1-1-onto→x ↔ g:A–1-1-onto→x))
17		f1oeq3 3677	. . . . . 6 ⊢ (x = A → (g:A–1-1-onto→x ↔ g:A–1-1-onto→A))
18	16, 17	bitrd 527	. . . . 5 ⊢ (x = A → (g:x–1-1-onto→x ↔ g:A–1-1-onto→A))
19	15, 18	3anbi12d 892	. . . 4 ⊢ (x = A → ((f:x–1-1-onto→x ⋀ g:x–1-1-onto→x ⋀ h = (f ∘ g)) ↔ (f:A–1-1-onto→A ⋀ g:A–1-1-onto→A ⋀ h = (f ∘ g))))
20	19	oprabbidv 3987	. . 3 ⊢ (x = A → {⟨⟨f, g⟩, h⟩∣(f:x–1-1-onto→x ⋀ g:x–1-1-onto→x ⋀ h = (f ∘ g))} = {⟨⟨f, g⟩, h⟩∣(f:A–1-1-onto→A ⋀ g:A–1-1-onto→A ⋀ h = (f ∘ g))})
21	3, 12, 20	fvopab 3781	. 2 ⊢ ({⟨x, y⟩∣y = {⟨⟨f, g⟩, h⟩∣(f:x–1-1-onto→x ⋀ g:x–1-1-onto→x ⋀ h = (f ∘ g))}} ‘A) = {⟨⟨f, g⟩, h⟩∣(f:A–1-1-onto→A ⋀ g:A–1-1-onto→A ⋀ h = (f ∘ g))}
22	2, 21, 11	3eqtr 1496	1 ⊢ (SymGrp ‘A) = {⟨⟨f, g⟩, h⟩∣((f ∈ P ⋀ g ∈ P) ⋀ h = (f ∘ g))}

Syntax hints:   ⋀ wa 223   ⋀ w3a 774   = wceq 954   ∈ wcel 956  {cab 1461  Vcvv 1807  {copab 2661   ∘ ccom 3169  –1-1-onto→wf1o 3176   ‘cfv 3177  {copab2 3955  SymGrpcsymgrp 10333

This theorem is referenced by:  symgoprval 10338  symgf 10339

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-rep 2688  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-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-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-id 2830  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-fo 3191  df-f1o 3192  df-fv 3193  df-oprab 3957  df-symgrp 10334

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