Theorem symgfixf1 19374

Description: The mapping of a permutation of a set fixing an element to a permutation of the set without the fixed element is a 1-1 function. (Contributed by AV, 4-Jan-2019.)

Hypotheses

Ref	Expression
symgfixf.p	⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
symgfixf.q	⊢ 𝑄 = {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}
symgfixf.s	⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))
symgfixf.h	⊢ 𝐻 = (𝑞 ∈ 𝑄 ↦ (𝑞 ↾ (𝑁 ∖ {𝐾})))

Assertion

Ref	Expression
symgfixf1	⊢ (𝐾 ∈ 𝑁 → 𝐻:𝑄–1-1→𝑆)

Distinct variable groups:   𝐾,𝑞   𝑃,𝑞   𝑁,𝑞   𝑄,𝑞   𝑆,𝑞

Allowed substitution hint:   𝐻(𝑞)

Proof of Theorem symgfixf1

Dummy variables 𝑔 𝑝 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		symgfixf.p	. . 3 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
2		symgfixf.q	. . 3 ⊢ 𝑄 = {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}
3		symgfixf.s	. . 3 ⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))
4		symgfixf.h	. . 3 ⊢ 𝐻 = (𝑞 ∈ 𝑄 ↦ (𝑞 ↾ (𝑁 ∖ {𝐾})))
5	1, 2, 3, 4	symgfixf 19373	. 2 ⊢ (𝐾 ∈ 𝑁 → 𝐻:𝑄⟶𝑆)
6	4	fvtresfn 6973	. . . . . 6 ⊢ (𝑔 ∈ 𝑄 → (𝐻‘𝑔) = (𝑔 ↾ (𝑁 ∖ {𝐾})))
7	4	fvtresfn 6973	. . . . . 6 ⊢ (𝑝 ∈ 𝑄 → (𝐻‘𝑝) = (𝑝 ↾ (𝑁 ∖ {𝐾})))
8	6, 7	eqeqan12d 2744	. . . . 5 ⊢ ((𝑔 ∈ 𝑄 ∧ 𝑝 ∈ 𝑄) → ((𝐻‘𝑔) = (𝐻‘𝑝) ↔ (𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾}))))
9	8	adantl 481	. . . 4 ⊢ ((𝐾 ∈ 𝑁 ∧ (𝑔 ∈ 𝑄 ∧ 𝑝 ∈ 𝑄)) → ((𝐻‘𝑔) = (𝐻‘𝑝) ↔ (𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾}))))
10	1, 2	symgfixelq 19370	. . . . . . 7 ⊢ (𝑔 ∈ V → (𝑔 ∈ 𝑄 ↔ (𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾)))
11	10	elv 3455	. . . . . 6 ⊢ (𝑔 ∈ 𝑄 ↔ (𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾))
12	1, 2	symgfixelq 19370	. . . . . . 7 ⊢ (𝑝 ∈ V → (𝑝 ∈ 𝑄 ↔ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾)))
13	12	elv 3455	. . . . . 6 ⊢ (𝑝 ∈ 𝑄 ↔ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾))
14	11, 13	anbi12i 628	. . . . 5 ⊢ ((𝑔 ∈ 𝑄 ∧ 𝑝 ∈ 𝑄) ↔ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾)))
15		f1ofn 6804	. . . . . . . . . . 11 ⊢ (𝑔:𝑁–1-1-onto→𝑁 → 𝑔 Fn 𝑁)
16	15	adantr 480	. . . . . . . . . 10 ⊢ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) → 𝑔 Fn 𝑁)
17		f1ofn 6804	. . . . . . . . . . 11 ⊢ (𝑝:𝑁–1-1-onto→𝑁 → 𝑝 Fn 𝑁)
18	17	adantr 480	. . . . . . . . . 10 ⊢ ((𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾) → 𝑝 Fn 𝑁)
19	16, 18	anim12i 613	. . . . . . . . 9 ⊢ (((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾)) → (𝑔 Fn 𝑁 ∧ 𝑝 Fn 𝑁))
20		difss 4102	. . . . . . . . 9 ⊢ (𝑁 ∖ {𝐾}) ⊆ 𝑁
21	19, 20	jctir 520	. . . . . . . 8 ⊢ (((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾)) → ((𝑔 Fn 𝑁 ∧ 𝑝 Fn 𝑁) ∧ (𝑁 ∖ {𝐾}) ⊆ 𝑁))
22	21	adantl 481	. . . . . . 7 ⊢ ((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾))) → ((𝑔 Fn 𝑁 ∧ 𝑝 Fn 𝑁) ∧ (𝑁 ∖ {𝐾}) ⊆ 𝑁))
23		fvreseq 7015	. . . . . . 7 ⊢ (((𝑔 Fn 𝑁 ∧ 𝑝 Fn 𝑁) ∧ (𝑁 ∖ {𝐾}) ⊆ 𝑁) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) ↔ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)))
24	22, 23	syl 17	. . . . . 6 ⊢ ((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾))) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) ↔ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)))
25		f1of 6803	. . . . . . . . . . . 12 ⊢ (𝑔:𝑁–1-1-onto→𝑁 → 𝑔:𝑁⟶𝑁)
26	25	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) → 𝑔:𝑁⟶𝑁)
27		f1of 6803	. . . . . . . . . . . 12 ⊢ (𝑝:𝑁–1-1-onto→𝑁 → 𝑝:𝑁⟶𝑁)
28	27	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾) → 𝑝:𝑁⟶𝑁)
29		fdm 6700	. . . . . . . . . . . 12 ⊢ (𝑔:𝑁⟶𝑁 → dom 𝑔 = 𝑁)
30		fdm 6700	. . . . . . . . . . . 12 ⊢ (𝑝:𝑁⟶𝑁 → dom 𝑝 = 𝑁)
31	29, 30	anim12i 613	. . . . . . . . . . 11 ⊢ ((𝑔:𝑁⟶𝑁 ∧ 𝑝:𝑁⟶𝑁) → (dom 𝑔 = 𝑁 ∧ dom 𝑝 = 𝑁))
32	26, 28, 31	syl2an 596	. . . . . . . . . 10 ⊢ (((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾)) → (dom 𝑔 = 𝑁 ∧ dom 𝑝 = 𝑁))
33		eqtr3 2752	. . . . . . . . . 10 ⊢ ((dom 𝑔 = 𝑁 ∧ dom 𝑝 = 𝑁) → dom 𝑔 = dom 𝑝)
34	32, 33	syl 17	. . . . . . . . 9 ⊢ (((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾)) → dom 𝑔 = dom 𝑝)
35	34	ad2antlr 727	. . . . . . . 8 ⊢ (((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)) → dom 𝑔 = dom 𝑝)
36		simpr 484	. . . . . . . . . 10 ⊢ (((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)) → ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖))
37		eqtr3 2752	. . . . . . . . . . . 12 ⊢ (((𝑔‘𝐾) = 𝐾 ∧ (𝑝‘𝐾) = 𝐾) → (𝑔‘𝐾) = (𝑝‘𝐾))
38	37	ad2ant2l 746	. . . . . . . . . . 11 ⊢ (((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾)) → (𝑔‘𝐾) = (𝑝‘𝐾))
39	38	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)) → (𝑔‘𝐾) = (𝑝‘𝐾))
40		fveq2 6861	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐾 → (𝑔‘𝑖) = (𝑔‘𝐾))
41		fveq2 6861	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐾 → (𝑝‘𝑖) = (𝑝‘𝐾))
42	40, 41	eqeq12d 2746	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐾 → ((𝑔‘𝑖) = (𝑝‘𝑖) ↔ (𝑔‘𝐾) = (𝑝‘𝐾)))
43	42	ralunsn 4861	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ 𝑁 → (∀𝑖 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖) ↔ (∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖) ∧ (𝑔‘𝐾) = (𝑝‘𝐾))))
44	43	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾))) → (∀𝑖 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖) ↔ (∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖) ∧ (𝑔‘𝐾) = (𝑝‘𝐾))))
45	44	adantr 480	. . . . . . . . . 10 ⊢ (((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)) → (∀𝑖 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖) ↔ (∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖) ∧ (𝑔‘𝐾) = (𝑝‘𝐾))))
46	36, 39, 45	mpbir2and 713	. . . . . . . . 9 ⊢ (((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)) → ∀𝑖 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖))
47		f1odm 6807	. . . . . . . . . . . . 13 ⊢ (𝑔:𝑁–1-1-onto→𝑁 → dom 𝑔 = 𝑁)
48	47	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) → dom 𝑔 = 𝑁)
49	48	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾)) → dom 𝑔 = 𝑁)
50		difsnid 4777	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ 𝑁 → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) = 𝑁)
51	50	eqcomd 2736	. . . . . . . . . . 11 ⊢ (𝐾 ∈ 𝑁 → 𝑁 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
52	49, 51	sylan9eqr 2787	. . . . . . . . . 10 ⊢ ((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾))) → dom 𝑔 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
53	52	adantr 480	. . . . . . . . 9 ⊢ (((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)) → dom 𝑔 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
54	46, 53	raleqtrrdv 3305	. . . . . . . 8 ⊢ (((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)) → ∀𝑖 ∈ dom 𝑔(𝑔‘𝑖) = (𝑝‘𝑖))
55		f1ofun 6805	. . . . . . . . . . . 12 ⊢ (𝑔:𝑁–1-1-onto→𝑁 → Fun 𝑔)
56	55	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) → Fun 𝑔)
57		f1ofun 6805	. . . . . . . . . . . 12 ⊢ (𝑝:𝑁–1-1-onto→𝑁 → Fun 𝑝)
58	57	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾) → Fun 𝑝)
59	56, 58	anim12i 613	. . . . . . . . . 10 ⊢ (((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾)) → (Fun 𝑔 ∧ Fun 𝑝))
60	59	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)) → (Fun 𝑔 ∧ Fun 𝑝))
61		eqfunfv 7011	. . . . . . . . 9 ⊢ ((Fun 𝑔 ∧ Fun 𝑝) → (𝑔 = 𝑝 ↔ (dom 𝑔 = dom 𝑝 ∧ ∀𝑖 ∈ dom 𝑔(𝑔‘𝑖) = (𝑝‘𝑖))))
62	60, 61	syl 17	. . . . . . . 8 ⊢ (((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)) → (𝑔 = 𝑝 ↔ (dom 𝑔 = dom 𝑝 ∧ ∀𝑖 ∈ dom 𝑔(𝑔‘𝑖) = (𝑝‘𝑖))))
63	35, 54, 62	mpbir2and 713	. . . . . . 7 ⊢ (((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)) → 𝑔 = 𝑝)
64	63	ex 412	. . . . . 6 ⊢ ((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾))) → (∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖) → 𝑔 = 𝑝))
65	24, 64	sylbid 240	. . . . 5 ⊢ ((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾))) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) → 𝑔 = 𝑝))
66	14, 65	sylan2b 594	. . . 4 ⊢ ((𝐾 ∈ 𝑁 ∧ (𝑔 ∈ 𝑄 ∧ 𝑝 ∈ 𝑄)) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) → 𝑔 = 𝑝))
67	9, 66	sylbid 240	. . 3 ⊢ ((𝐾 ∈ 𝑁 ∧ (𝑔 ∈ 𝑄 ∧ 𝑝 ∈ 𝑄)) → ((𝐻‘𝑔) = (𝐻‘𝑝) → 𝑔 = 𝑝))
68	67	ralrimivva 3181	. 2 ⊢ (𝐾 ∈ 𝑁 → ∀𝑔 ∈ 𝑄 ∀𝑝 ∈ 𝑄 ((𝐻‘𝑔) = (𝐻‘𝑝) → 𝑔 = 𝑝))
69		dff13 7232	. 2 ⊢ (𝐻:𝑄–1-1→𝑆 ↔ (𝐻:𝑄⟶𝑆 ∧ ∀𝑔 ∈ 𝑄 ∀𝑝 ∈ 𝑄 ((𝐻‘𝑔) = (𝐻‘𝑝) → 𝑔 = 𝑝)))
70	5, 68, 69	sylanbrc 583	1 ⊢ (𝐾 ∈ 𝑁 → 𝐻:𝑄–1-1→𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  {crab 3408  Vcvv 3450   ∖ cdif 3914   ∪ cun 3915   ⊆ wss 3917  {csn 4592   ↦ cmpt 5191  dom cdm 5641   ↾ cres 5643  Fun wfun 6508   Fn wfn 6509  ⟶wf 6510  –1-1→wf1 6511  –1-1-onto→wf1o 6513  ‘cfv 6514  Basecbs 17186  SymGrpcsymg 19306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-uz 12801  df-fz 13476  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-tset 17246  df-efmnd 18803  df-symg 19307

This theorem is referenced by:  symgfixf1o  19377