Theorem symgfixf1 19304

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 19303	. 2 ⊢ (𝐾 ∈ 𝑁 → 𝐻:𝑄⟶𝑆)
6	4	fvtresfn 7000	. . . . . 6 ⊢ (𝑔 ∈ 𝑄 → (𝐻‘𝑔) = (𝑔 ↾ (𝑁 ∖ {𝐾})))
7	4	fvtresfn 7000	. . . . . 6 ⊢ (𝑝 ∈ 𝑄 → (𝐻‘𝑝) = (𝑝 ↾ (𝑁 ∖ {𝐾})))
8	6, 7	eqeqan12d 2746	. . . . 5 ⊢ ((𝑔 ∈ 𝑄 ∧ 𝑝 ∈ 𝑄) → ((𝐻‘𝑔) = (𝐻‘𝑝) ↔ (𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾}))))
9	8	adantl 482	. . . 4 ⊢ ((𝐾 ∈ 𝑁 ∧ (𝑔 ∈ 𝑄 ∧ 𝑝 ∈ 𝑄)) → ((𝐻‘𝑔) = (𝐻‘𝑝) ↔ (𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾}))))
10	1, 2	symgfixelq 19300	. . . . . . 7 ⊢ (𝑔 ∈ V → (𝑔 ∈ 𝑄 ↔ (𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾)))
11	10	elv 3480	. . . . . 6 ⊢ (𝑔 ∈ 𝑄 ↔ (𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾))
12	1, 2	symgfixelq 19300	. . . . . . 7 ⊢ (𝑝 ∈ V → (𝑝 ∈ 𝑄 ↔ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾)))
13	12	elv 3480	. . . . . 6 ⊢ (𝑝 ∈ 𝑄 ↔ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾))
14	11, 13	anbi12i 627	. . . . 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 6834	. . . . . . . . . . 11 ⊢ (𝑔:𝑁–1-1-onto→𝑁 → 𝑔 Fn 𝑁)
16	15	adantr 481	. . . . . . . . . 10 ⊢ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) → 𝑔 Fn 𝑁)
17		f1ofn 6834	. . . . . . . . . . 11 ⊢ (𝑝:𝑁–1-1-onto→𝑁 → 𝑝 Fn 𝑁)
18	17	adantr 481	. . . . . . . . . 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 4131	. . . . . . . . 9 ⊢ (𝑁 ∖ {𝐾}) ⊆ 𝑁
21	19, 20	jctir 521	. . . . . . . 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 482	. . . . . . 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 7041	. . . . . . 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 6833	. . . . . . . . . . . 12 ⊢ (𝑔:𝑁–1-1-onto→𝑁 → 𝑔:𝑁⟶𝑁)
26	25	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) → 𝑔:𝑁⟶𝑁)
27		f1of 6833	. . . . . . . . . . . 12 ⊢ (𝑝:𝑁–1-1-onto→𝑁 → 𝑝:𝑁⟶𝑁)
28	27	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾) → 𝑝:𝑁⟶𝑁)
29		fdm 6726	. . . . . . . . . . . 12 ⊢ (𝑔:𝑁⟶𝑁 → dom 𝑔 = 𝑁)
30		fdm 6726	. . . . . . . . . . . 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 2758	. . . . . . . . . 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 725	. . . . . . . 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 485	. . . . . . . . . 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 2758	. . . . . . . . . . . 12 ⊢ (((𝑔‘𝐾) = 𝐾 ∧ (𝑝‘𝐾) = 𝐾) → (𝑔‘𝐾) = (𝑝‘𝐾))
38	37	ad2ant2l 744	. . . . . . . . . . 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 725	. . . . . . . . . 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 6891	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐾 → (𝑔‘𝑖) = (𝑔‘𝐾))
41		fveq2 6891	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐾 → (𝑝‘𝑖) = (𝑝‘𝐾))
42	40, 41	eqeq12d 2748	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐾 → ((𝑔‘𝑖) = (𝑝‘𝑖) ↔ (𝑔‘𝐾) = (𝑝‘𝐾)))
43	42	ralunsn 4894	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 481	. . . . . . . . . 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 711	. . . . . . . . 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 6837	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝑁–1-1-onto→𝑁 → dom 𝑔 = 𝑁)
48	47	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) → dom 𝑔 = 𝑁)
49	48	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾)) → dom 𝑔 = 𝑁)
50		difsnid 4813	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ 𝑁 → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) = 𝑁)
51	50	eqcomd 2738	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ 𝑁 → 𝑁 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
52	49, 51	sylan9eqr 2794	. . . . . . . . . . 11 ⊢ ((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 481	. . . . . . . . . 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)) → dom 𝑔 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
54	53	raleqdv 3325	. . . . . . . . 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 ∈ dom 𝑔(𝑔‘𝑖) = (𝑝‘𝑖) ↔ ∀𝑖 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)))
55	46, 54	mpbird 256	. . . . . . . 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))
56		f1ofun 6835	. . . . . . . . . . . 12 ⊢ (𝑔:𝑁–1-1-onto→𝑁 → Fun 𝑔)
57	56	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) → Fun 𝑔)
58		f1ofun 6835	. . . . . . . . . . . 12 ⊢ (𝑝:𝑁–1-1-onto→𝑁 → Fun 𝑝)
59	58	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾) → Fun 𝑝)
60	57, 59	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))
61	60	ad2antlr 725	. . . . . . . . 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))
62		eqfunfv 7037	. . . . . . . . 9 ⊢ ((Fun 𝑔 ∧ Fun 𝑝) → (𝑔 = 𝑝 ↔ (dom 𝑔 = dom 𝑝 ∧ ∀𝑖 ∈ dom 𝑔(𝑔‘𝑖) = (𝑝‘𝑖))))
63	61, 62	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))))
64	35, 55, 63	mpbir2and 711	. . . . . . 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)
65	64	ex 413	. . . . . 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))
66	24, 65	sylbid 239	. . . . 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))
67	14, 66	sylan2b 594	. . . 4 ⊢ ((𝐾 ∈ 𝑁 ∧ (𝑔 ∈ 𝑄 ∧ 𝑝 ∈ 𝑄)) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) → 𝑔 = 𝑝))
68	9, 67	sylbid 239	. . 3 ⊢ ((𝐾 ∈ 𝑁 ∧ (𝑔 ∈ 𝑄 ∧ 𝑝 ∈ 𝑄)) → ((𝐻‘𝑔) = (𝐻‘𝑝) → 𝑔 = 𝑝))
69	68	ralrimivva 3200	. 2 ⊢ (𝐾 ∈ 𝑁 → ∀𝑔 ∈ 𝑄 ∀𝑝 ∈ 𝑄 ((𝐻‘𝑔) = (𝐻‘𝑝) → 𝑔 = 𝑝))
70		dff13 7253	. 2 ⊢ (𝐻:𝑄–1-1→𝑆 ↔ (𝐻:𝑄⟶𝑆 ∧ ∀𝑔 ∈ 𝑄 ∀𝑝 ∈ 𝑄 ((𝐻‘𝑔) = (𝐻‘𝑝) → 𝑔 = 𝑝)))
71	5, 69, 70	sylanbrc 583	1 ⊢ (𝐾 ∈ 𝑁 → 𝐻:𝑄–1-1→𝑆)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  {crab 3432  Vcvv 3474   ∖ cdif 3945   ∪ cun 3946   ⊆ wss 3948  {csn 4628   ↦ cmpt 5231  dom cdm 5676   ↾ cres 5678  Fun wfun 6537   Fn wfn 6538  ⟶wf 6539  –1-1→wf1 6540  –1-1-onto→wf1o 6542  ‘cfv 6543  Basecbs 17143  SymGrpcsymg 19233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12472  df-z 12558  df-uz 12822  df-fz 13484  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-tset 17215  df-efmnd 18749  df-symg 19234

This theorem is referenced by:  symgfixf1o  19307