Theorem symgfixf1 17857

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 17856	. 2 ⊢ (𝐾 ∈ 𝑁 → 𝐻:𝑄⟶𝑆)
6	4	fvtresfn 6284	. . . . . 6 ⊢ (𝑔 ∈ 𝑄 → (𝐻‘𝑔) = (𝑔 ↾ (𝑁 ∖ {𝐾})))
7	4	fvtresfn 6284	. . . . . 6 ⊢ (𝑝 ∈ 𝑄 → (𝐻‘𝑝) = (𝑝 ↾ (𝑁 ∖ {𝐾})))
8	6, 7	eqeqan12d 2638	. . . . 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		vex 3203	. . . . . . 7 ⊢ 𝑔 ∈ V
11	1, 2	symgfixelq 17853	. . . . . . 7 ⊢ (𝑔 ∈ V → (𝑔 ∈ 𝑄 ↔ (𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾)))
12	10, 11	ax-mp 5	. . . . . 6 ⊢ (𝑔 ∈ 𝑄 ↔ (𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾))
13		vex 3203	. . . . . . 7 ⊢ 𝑝 ∈ V
14	1, 2	symgfixelq 17853	. . . . . . 7 ⊢ (𝑝 ∈ V → (𝑝 ∈ 𝑄 ↔ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾)))
15	13, 14	ax-mp 5	. . . . . 6 ⊢ (𝑝 ∈ 𝑄 ↔ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾))
16	12, 15	anbi12i 733	. . . . 5 ⊢ ((𝑔 ∈ 𝑄 ∧ 𝑝 ∈ 𝑄) ↔ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾)))
17		f1ofn 6138	. . . . . . . . . . 11 ⊢ (𝑔:𝑁–1-1-onto→𝑁 → 𝑔 Fn 𝑁)
18	17	adantr 481	. . . . . . . . . 10 ⊢ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) → 𝑔 Fn 𝑁)
19		f1ofn 6138	. . . . . . . . . . 11 ⊢ (𝑝:𝑁–1-1-onto→𝑁 → 𝑝 Fn 𝑁)
20	19	adantr 481	. . . . . . . . . 10 ⊢ ((𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾) → 𝑝 Fn 𝑁)
21	18, 20	anim12i 590	. . . . . . . . 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))
22		difss 3737	. . . . . . . . 9 ⊢ (𝑁 ∖ {𝐾}) ⊆ 𝑁
23	21, 22	jctir 561	. . . . . . . 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))
24	23	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))
25		fvreseq 6319	. . . . . . 7 ⊢ (((𝑔 Fn 𝑁 ∧ 𝑝 Fn 𝑁) ∧ (𝑁 ∖ {𝐾}) ⊆ 𝑁) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) ↔ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖)))
26	24, 25	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)))
27		f1of 6137	. . . . . . . . . . . 12 ⊢ (𝑔:𝑁–1-1-onto→𝑁 → 𝑔:𝑁⟶𝑁)
28	27	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) → 𝑔:𝑁⟶𝑁)
29		f1of 6137	. . . . . . . . . . . 12 ⊢ (𝑝:𝑁–1-1-onto→𝑁 → 𝑝:𝑁⟶𝑁)
30	29	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾) → 𝑝:𝑁⟶𝑁)
31		fdm 6051	. . . . . . . . . . . 12 ⊢ (𝑔:𝑁⟶𝑁 → dom 𝑔 = 𝑁)
32		fdm 6051	. . . . . . . . . . . 12 ⊢ (𝑝:𝑁⟶𝑁 → dom 𝑝 = 𝑁)
33	31, 32	anim12i 590	. . . . . . . . . . 11 ⊢ ((𝑔:𝑁⟶𝑁 ∧ 𝑝:𝑁⟶𝑁) → (dom 𝑔 = 𝑁 ∧ dom 𝑝 = 𝑁))
34	28, 30, 33	syl2an 494	. . . . . . . . . 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))
35		eqtr3 2643	. . . . . . . . . 10 ⊢ ((dom 𝑔 = 𝑁 ∧ dom 𝑝 = 𝑁) → dom 𝑔 = dom 𝑝)
36	34, 35	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)
37	36	ad2antlr 763	. . . . . . . 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)
38		simpr 477	. . . . . . . . . 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))
39		eqtr3 2643	. . . . . . . . . . . 12 ⊢ (((𝑔‘𝐾) = 𝐾 ∧ (𝑝‘𝐾) = 𝐾) → (𝑔‘𝐾) = (𝑝‘𝐾))
40	39	ad2ant2l 782	. . . . . . . . . . 11 ⊢ (((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾)) → (𝑔‘𝐾) = (𝑝‘𝐾))
41	40	ad2antlr 763	. . . . . . . . . 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))
42		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐾 → (𝑔‘𝑖) = (𝑔‘𝐾))
43		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐾 → (𝑝‘𝑖) = (𝑝‘𝐾))
44	42, 43	eqeq12d 2637	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐾 → ((𝑔‘𝑖) = (𝑝‘𝑖) ↔ (𝑔‘𝐾) = (𝑝‘𝐾)))
45	44	ralunsn 4422	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ 𝑁 → (∀𝑖 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖) ↔ (∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔‘𝑖) = (𝑝‘𝑖) ∧ (𝑔‘𝐾) = (𝑝‘𝐾))))
46	45	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))))
47	46	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))))
48	38, 41, 47	mpbir2and 957	. . . . . . . . 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))
49		f1odm 6141	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝑁–1-1-onto→𝑁 → dom 𝑔 = 𝑁)
50	49	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) → dom 𝑔 = 𝑁)
51	50	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾)) → dom 𝑔 = 𝑁)
52		difsnid 4341	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ 𝑁 → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) = 𝑁)
53	52	eqcomd 2628	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ 𝑁 → 𝑁 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
54	51, 53	sylan9eqr 2678	. . . . . . . . . . 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}))
55	54	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}))
56	55	raleqdv 3144	. . . . . . . . 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)))
57	48, 56	mpbird 247	. . . . . . . 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))
58		f1ofun 6139	. . . . . . . . . . . 12 ⊢ (𝑔:𝑁–1-1-onto→𝑁 → Fun 𝑔)
59	58	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) → Fun 𝑔)
60		f1ofun 6139	. . . . . . . . . . . 12 ⊢ (𝑝:𝑁–1-1-onto→𝑁 → Fun 𝑝)
61	60	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾) → Fun 𝑝)
62	59, 61	anim12i 590	. . . . . . . . . 10 ⊢ (((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾)) → (Fun 𝑔 ∧ Fun 𝑝))
63	62	ad2antlr 763	. . . . . . . . 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))
64		eqfunfv 6316	. . . . . . . . 9 ⊢ ((Fun 𝑔 ∧ Fun 𝑝) → (𝑔 = 𝑝 ↔ (dom 𝑔 = dom 𝑝 ∧ ∀𝑖 ∈ dom 𝑔(𝑔‘𝑖) = (𝑝‘𝑖))))
65	63, 64	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))))
66	37, 57, 65	mpbir2and 957	. . . . . . 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)
67	66	ex 450	. . . . . 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))
68	26, 67	sylbid 230	. . . . 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))
69	16, 68	sylan2b 492	. . . 4 ⊢ ((𝐾 ∈ 𝑁 ∧ (𝑔 ∈ 𝑄 ∧ 𝑝 ∈ 𝑄)) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) → 𝑔 = 𝑝))
70	9, 69	sylbid 230	. . 3 ⊢ ((𝐾 ∈ 𝑁 ∧ (𝑔 ∈ 𝑄 ∧ 𝑝 ∈ 𝑄)) → ((𝐻‘𝑔) = (𝐻‘𝑝) → 𝑔 = 𝑝))
71	70	ralrimivva 2971	. 2 ⊢ (𝐾 ∈ 𝑁 → ∀𝑔 ∈ 𝑄 ∀𝑝 ∈ 𝑄 ((𝐻‘𝑔) = (𝐻‘𝑝) → 𝑔 = 𝑝))
72		dff13 6512	. 2 ⊢ (𝐻:𝑄–1-1→𝑆 ↔ (𝐻:𝑄⟶𝑆 ∧ ∀𝑔 ∈ 𝑄 ∀𝑝 ∈ 𝑄 ((𝐻‘𝑔) = (𝐻‘𝑝) → 𝑔 = 𝑝)))
73	5, 71, 72	sylanbrc 698	1 ⊢ (𝐾 ∈ 𝑁 → 𝐻:𝑄–1-1→𝑆)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ⊆ wss 3574  {csn 4177   ↦ cmpt 4729  dom cdm 5114   ↾ cres 5116  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  –1-1→wf1 5885  –1-1-onto→wf1o 5887  ‘cfv 5888  Basecbs 15857  SymGrpcsymg 17797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-tset 15960  df-symg 17798

This theorem is referenced by:  symgfixf1o  17860