Theorem symgfixf1 19702

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⟶1-1𝑆)

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 19701	. 2 ⊢ (𝐾 ∈ 𝑁 → 𝐻:𝑄⟶𝑆)
6	4	fvtresfn 7027	. . . . . 6 ⊢ (𝑔 ∈ 𝑄 → (𝐻‘𝑔) = (𝑔 ↾ (𝑁 ∖ {𝐾})))
7	4	fvtresfn 7027	. . . . . 6 ⊢ (𝑝 ∈ 𝑄 → (𝐻‘𝑝) = (𝑝 ↾ (𝑁 ∖ {𝐾})))
8	6, 7	eqeqan12d 2777	. . . . 5 ⊢ ((𝑔 ∈ 𝑄 ∧ 𝑝 ∈ 𝑄) → ((𝐻‘𝑔) = (𝐻‘𝑝) ↔ (𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾}))))
9	8	adantl 487	. . . 4 ⊢ ((𝐾 ∈ 𝑁 ∧ (𝑔 ∈ 𝑄 ∧ 𝑝 ∈ 𝑄)) → ((𝐻‘𝑔) = (𝐻‘𝑝) ↔ (𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾}))))
10	1, 2	symgfixelq 19698	. . . . . . 7 ⊢ (𝑔 ∈ V → (𝑔 ∈ 𝑄 ↔ (𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾)))
11	10	elv 3459	. . . . . 6 ⊢ (𝑔 ∈ 𝑄 ↔ (𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾))
12	1, 2	symgfixelq 19698	. . . . . . 7 ⊢ (𝑝 ∈ V → (𝑝 ∈ 𝑄 ↔ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾)))
13	12	elv 3459	. . . . . 6 ⊢ (𝑝 ∈ 𝑄 ↔ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾))
14	11, 13	anbi12i 640	. . . . 5 ⊢ ((𝑔 ∈ 𝑄 ∧ 𝑝 ∈ 𝑄) ↔ ((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾)))
15		f1ofn 6855	. . . . . . . . . . 11 ⊢ (𝑔:𝑁⟶1-1onto𝑁 → 𝑔 Fn 𝑁)
16	15	adantr 486	. . . . . . . . . 10 ⊢ ((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) → 𝑔 Fn 𝑁)
17		f1ofn 6855	. . . . . . . . . . 11 ⊢ (𝑝:𝑁⟶1-1onto𝑁 → 𝑝 Fn 𝑁)
18	17	adantr 486	. . . . . . . . . 10 ⊢ ((𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾) → 𝑝 Fn 𝑁)
19	16, 18	anim12i 625	. . . . . . . . 9 ⊢ (((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾)) → (𝑔 Fn 𝑁 ∧ 𝑝 Fn 𝑁))
20		difss 4075	. . . . . . . . 9 ⊢ (𝑁 ∖ {𝐾}) ⊆ 𝑁
21	19, 20	jctir 530	. . . . . . . 8 ⊢ (((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾)) → ((𝑔 Fn 𝑁 ∧ 𝑝 Fn 𝑁) ∧ (𝑁 ∖ {𝐾}) ⊆ 𝑁))
22	21	adantl 487	. . . . . . 7 ⊢ ((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾))) → ((𝑔 Fn 𝑁 ∧ 𝑝 Fn 𝑁) ∧ (𝑁 ∖ {𝐾}) ⊆ 𝑁))
23		fvreseq 7070	. . . . . . 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 18	. . . . . 6 ⊢ ((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾))) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) ↔ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)))
25		f1of 6854	. . . . . . . . . . . 12 ⊢ (𝑔:𝑁⟶1-1onto𝑁 → 𝑔:𝑁⟶𝑁)
26	25	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) → 𝑔:𝑁⟶𝑁)
27		f1of 6854	. . . . . . . . . . . 12 ⊢ (𝑝:𝑁⟶1-1onto𝑁 → 𝑝:𝑁⟶𝑁)
28	27	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾) → 𝑝:𝑁⟶𝑁)
29		fdm 6749	. . . . . . . . . . . 12 ⊢ (𝑔:𝑁⟶𝑁 → dom 𝑔 = 𝑁)
30		fdm 6749	. . . . . . . . . . . 12 ⊢ (𝑝:𝑁⟶𝑁 → dom 𝑝 = 𝑁)
31	29, 30	anim12i 625	. . . . . . . . . . 11 ⊢ ((𝑔:𝑁⟶𝑁 ∧ 𝑝:𝑁⟶𝑁) → (dom 𝑔 = 𝑁 ∧ dom 𝑝 = 𝑁))
32	26, 28, 31	syl2an 608	. . . . . . . . . 10 ⊢ (((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾)) → (dom 𝑔 = 𝑁 ∧ dom 𝑝 = 𝑁))
33		eqtr3 2785	. . . . . . . . . 10 ⊢ ((dom 𝑔 = 𝑁 ∧ dom 𝑝 = 𝑁) → dom 𝑔 = dom 𝑝)
34	32, 33	syl 18	. . . . . . . . 9 ⊢ (((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾)) → dom 𝑔 = dom 𝑝)
35	34	ad2antlr 740	. . . . . . . 8 ⊢ (((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)) → dom 𝑔 = dom 𝑝)
36		simpr 490	. . . . . . . . . 10 ⊢ (((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)) → ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖))
37		eqtr3 2785	. . . . . . . . . . . 12 ⊢ (((𝑔‘𝐾) = 𝐾 ∧ (𝑝‘𝐾) = 𝐾) → (𝑔‘𝐾) = (𝑝‘𝐾))
38	37	ad2ant2l 759	. . . . . . . . . . 11 ⊢ (((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾)) → (𝑔‘𝐾) = (𝑝‘𝐾))
39	38	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)) → (𝑔‘𝐾) = (𝑝‘𝐾))
40		fveq2 6916	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐾 → (𝑔‘𝑖) = (𝑔‘𝐾))
41		fveq2 6916	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐾 → (𝑝‘𝑖) = (𝑝‘𝐾))
42	40, 41	eqeq12d 2779	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐾 → ((𝑔‘𝑖) = (𝑝‘𝑖) ↔ (𝑔‘𝐾) = (𝑝‘𝐾)))
43	42	ralunsn 4830	. . . . . . . . . . . 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 486	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (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 486	. . . . . . . . . 10 ⊢ (((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (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 726	. . . . . . . . 9 ⊢ (((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)) → ∀𝑖 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖))
47		f1odm 6858	. . . . . . . . . . . . 13 ⊢ (𝑔:𝑁⟶1-1onto𝑁 → dom 𝑔 = 𝑁)
48	47	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) → dom 𝑔 = 𝑁)
49	48	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾)) → dom 𝑔 = 𝑁)
50		difsnid 4773	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ 𝑁 → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) = 𝑁)
51	50	eqcomd 2769	. . . . . . . . . . 11 ⊢ (𝐾 ∈ 𝑁 → 𝑁 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
52	49, 51	sylan9eqr 2820	. . . . . . . . . 10 ⊢ ((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾))) → dom 𝑔 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
53	52	adantr 486	. . . . . . . . 9 ⊢ (((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)) → dom 𝑔 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
54	46, 53	raleqtrrdv 3327	. . . . . . . 8 ⊢ (((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)) → ∀𝑖 ∈ dom 𝑔(𝑔‘𝑖) = (𝑝‘𝑖))
55		f1ofun 6856	. . . . . . . . . . . 12 ⊢ (𝑔:𝑁⟶1-1onto𝑁 → Fun 𝑔)
56	55	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) → Fun 𝑔)
57		f1ofun 6856	. . . . . . . . . . . 12 ⊢ (𝑝:𝑁⟶1-1onto𝑁 → Fun 𝑝)
58	57	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾) → Fun 𝑝)
59	56, 58	anim12i 625	. . . . . . . . . 10 ⊢ (((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾)) → (Fun 𝑔 ∧ Fun 𝑝))
60	59	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)) → (Fun 𝑔 ∧ Fun 𝑝))
61		eqfunfv 7066	. . . . . . . . 9 ⊢ ((Fun 𝑔 ∧ Fun 𝑝) → (𝑔 = 𝑝 ↔ (dom 𝑔 = dom 𝑝 ∧ ∀𝑖 ∈ dom 𝑔(𝑔‘𝑖) = (𝑝‘𝑖))))
62	60, 61	syl 18	. . . . . . . 8 ⊢ (((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (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 726	. . . . . . 7 ⊢ (((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)) → 𝑔 = 𝑝)
64	63	ex 418	. . . . . 6 ⊢ ((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾))) → (∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖) → 𝑔 = 𝑝))
65	24, 64	sylbid 243	. . . . 5 ⊢ ((𝐾 ∈ 𝑁 ∧ ((𝑔:𝑁⟶1-1onto𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁⟶1-1onto𝑁 ∧ (𝑝‘𝐾) = 𝐾))) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) → 𝑔 = 𝑝))
66	14, 65	sylan2b 606	. . . 4 ⊢ ((𝐾 ∈ 𝑁 ∧ (𝑔 ∈ 𝑄 ∧ 𝑝 ∈ 𝑄)) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) → 𝑔 = 𝑝))
67	9, 66	sylbid 243	. . 3 ⊢ ((𝐾 ∈ 𝑁 ∧ (𝑔 ∈ 𝑄 ∧ 𝑝 ∈ 𝑄)) → ((𝐻‘𝑔) = (𝐻‘𝑝) → 𝑔 = 𝑝))
68	67	ralrimivva 3209	. 2 ⊢ (𝐾 ∈ 𝑁 → ∀𝑔 ∈ 𝑄 ∀𝑝 ∈ 𝑄 ((𝐻‘𝑔) = (𝐻‘𝑝) → 𝑔 = 𝑝))
69		dff13 7289	. 2 ⊢ (𝐻:𝑄⟶1-1𝑆 ↔ (𝐻:𝑄⟶𝑆 ∧ ∀𝑔 ∈ 𝑄 ∀𝑝 ∈ 𝑄 ((𝐻‘𝑔) = (𝐻‘𝑝) → 𝑔 = 𝑝)))
70	5, 68, 69	sylanbrc 595	1 ⊢ (𝐾 ∈ 𝑁 → 𝐻:𝑄⟶1-1𝑆)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   ∪ cun 4152  {csn 4588   ↦ cmpt 5192  dom cdm 5658   ↾ cres 5660  Fun wfun 6560   Fn wfn 6561  ⟶wf 6562  ⟶1-1wf1 6563  ⟶1-1ontowf1o 6565  ‘cfv 6566  Basecbs 17431  SymGrpcsymg 19634

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-z 12734  df-uz 13006  df-fz 13680  df-struct 17369  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-ress 17453  df-plusg 17486  df-tset 17492  df-efmnd 19115  df-symg 19635

This theorem is used by:  symgfixf1o  19705