Theorem symgfixf1 19367

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 19366	. 2 ⊢ (𝐾 ∈ 𝑁 → 𝐻:𝑄⟶𝑆)
6	4	fvtresfn 6970	. . . . . 6 ⊢ (𝑔 ∈ 𝑄 → (𝐻‘𝑔) = (𝑔 ↾ (𝑁 ∖ {𝐾})))
7	4	fvtresfn 6970	. . . . . 6 ⊢ (𝑝 ∈ 𝑄 → (𝐻‘𝑝) = (𝑝 ↾ (𝑁 ∖ {𝐾})))
8	6, 7	eqeqan12d 2743	. . . . 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 19363	. . . . . . 7 ⊢ (𝑔 ∈ V → (𝑔 ∈ 𝑄 ↔ (𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾)))
11	10	elv 3452	. . . . . 6 ⊢ (𝑔 ∈ 𝑄 ↔ (𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾))
12	1, 2	symgfixelq 19363	. . . . . . 7 ⊢ (𝑝 ∈ V → (𝑝 ∈ 𝑄 ↔ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾)))
13	12	elv 3452	. . . . . 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 6801	. . . . . . . . . . 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 6801	. . . . . . . . . . 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 4099	. . . . . . . . 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 7012	. . . . . . 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 6800	. . . . . . . . . . . 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 6800	. . . . . . . . . . . 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 6697	. . . . . . . . . . . 12 ⊢ (𝑔:𝑁⟶𝑁 → dom 𝑔 = 𝑁)
30		fdm 6697	. . . . . . . . . . . 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 2751	. . . . . . . . . 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 2751	. . . . . . . . . . . 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 6858	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐾 → (𝑔‘𝑖) = (𝑔‘𝐾))
41		fveq2 6858	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐾 → (𝑝‘𝑖) = (𝑝‘𝐾))
42	40, 41	eqeq12d 2745	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐾 → ((𝑔‘𝑖) = (𝑝‘𝑖) ↔ (𝑔‘𝐾) = (𝑝‘𝐾)))
43	42	ralunsn 4858	. . . . . . . . . . . 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 6804	. . . . . . . . . . . . 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 4774	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ 𝑁 → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) = 𝑁)
51	50	eqcomd 2735	. . . . . . . . . . 11 ⊢ (𝐾 ∈ 𝑁 → 𝑁 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
52	49, 51	sylan9eqr 2786	. . . . . . . . . 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 3303	. . . . . . . 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 6802	. . . . . . . . . . . 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 6802	. . . . . . . . . . . 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 7008	. . . . . . . . 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 3180	. 2 ⊢ (𝐾 ∈ 𝑁 → ∀𝑔 ∈ 𝑄 ∀𝑝 ∈ 𝑄 ((𝐻‘𝑔) = (𝐻‘𝑝) → 𝑔 = 𝑝))
69		dff13 7229	. 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 3044  {crab 3405  Vcvv 3447   ∖ cdif 3911   ∪ cun 3912   ⊆ wss 3914  {csn 4589   ↦ cmpt 5188  dom cdm 5638   ↾ cres 5640  Fun wfun 6505   Fn wfn 6506  ⟶wf 6507  –1-1→wf1 6508  –1-1-onto→wf1o 6510  ‘cfv 6511  Basecbs 17179  SymGrpcsymg 19299

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 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-uz 12794  df-fz 13469  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-tset 17239  df-efmnd 18796  df-symg 19300

This theorem is referenced by:  symgfixf1o  19370