Theorem symgfixf1 19305

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 19304	. 2 ⊢ (𝐾 ∈ 𝑁 → 𝐻:𝑄⟶𝑆)
6	4	fvtresfn 7001	. . . . . 6 ⊢ (𝑔 ∈ 𝑄 → (𝐻‘𝑔) = (𝑔 ↾ (𝑁 ∖ {𝐾})))
7	4	fvtresfn 7001	. . . . . 6 ⊢ (𝑝 ∈ 𝑄 → (𝐻‘𝑝) = (𝑝 ↾ (𝑁 ∖ {𝐾})))
8	6, 7	eqeqan12d 2747	. . . . 5 ⊢ ((𝑔 ∈ 𝑄 ∧ 𝑝 ∈ 𝑄) → ((𝐻‘𝑔) = (𝐻‘𝑝) ↔ (𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾}))))
9	8	adantl 483	. . . 4 ⊢ ((𝐾 ∈ 𝑁 ∧ (𝑔 ∈ 𝑄 ∧ 𝑝 ∈ 𝑄)) → ((𝐻‘𝑔) = (𝐻‘𝑝) ↔ (𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾}))))
10	1, 2	symgfixelq 19301	. . . . . . 7 ⊢ (𝑔 ∈ V → (𝑔 ∈ 𝑄 ↔ (𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾)))
11	10	elv 3481	. . . . . 6 ⊢ (𝑔 ∈ 𝑄 ↔ (𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾))
12	1, 2	symgfixelq 19301	. . . . . . 7 ⊢ (𝑝 ∈ V → (𝑝 ∈ 𝑄 ↔ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾)))
13	12	elv 3481	. . . . . 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 6835	. . . . . . . . . . 11 ⊢ (𝑔:𝑁–1-1-onto→𝑁 → 𝑔 Fn 𝑁)
16	15	adantr 482	. . . . . . . . . 10 ⊢ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) → 𝑔 Fn 𝑁)
17		f1ofn 6835	. . . . . . . . . . 11 ⊢ (𝑝:𝑁–1-1-onto→𝑁 → 𝑝 Fn 𝑁)
18	17	adantr 482	. . . . . . . . . 10 ⊢ ((𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾) → 𝑝 Fn 𝑁)
19	16, 18	anim12i 614	. . . . . . . . 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 4132	. . . . . . . . 9 ⊢ (𝑁 ∖ {𝐾}) ⊆ 𝑁
21	19, 20	jctir 522	. . . . . . . 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 483	. . . . . . 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 7042	. . . . . . 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 6834	. . . . . . . . . . . 12 ⊢ (𝑔:𝑁–1-1-onto→𝑁 → 𝑔:𝑁⟶𝑁)
26	25	adantr 482	. . . . . . . . . . 11 ⊢ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) → 𝑔:𝑁⟶𝑁)
27		f1of 6834	. . . . . . . . . . . 12 ⊢ (𝑝:𝑁–1-1-onto→𝑁 → 𝑝:𝑁⟶𝑁)
28	27	adantr 482	. . . . . . . . . . 11 ⊢ ((𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾) → 𝑝:𝑁⟶𝑁)
29		fdm 6727	. . . . . . . . . . . 12 ⊢ (𝑔:𝑁⟶𝑁 → dom 𝑔 = 𝑁)
30		fdm 6727	. . . . . . . . . . . 12 ⊢ (𝑝:𝑁⟶𝑁 → dom 𝑝 = 𝑁)
31	29, 30	anim12i 614	. . . . . . . . . . 11 ⊢ ((𝑔:𝑁⟶𝑁 ∧ 𝑝:𝑁⟶𝑁) → (dom 𝑔 = 𝑁 ∧ dom 𝑝 = 𝑁))
32	26, 28, 31	syl2an 597	. . . . . . . . . 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 2759	. . . . . . . . . 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 726	. . . . . . . 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 486	. . . . . . . . . 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 2759	. . . . . . . . . . . 12 ⊢ (((𝑔‘𝐾) = 𝐾 ∧ (𝑝‘𝐾) = 𝐾) → (𝑔‘𝐾) = (𝑝‘𝐾))
38	37	ad2ant2l 745	. . . . . . . . . . 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 726	. . . . . . . . . 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 6892	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐾 → (𝑔‘𝑖) = (𝑔‘𝐾))
41		fveq2 6892	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐾 → (𝑝‘𝑖) = (𝑝‘𝐾))
42	40, 41	eqeq12d 2749	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐾 → ((𝑔‘𝑖) = (𝑝‘𝑖) ↔ (𝑔‘𝐾) = (𝑝‘𝐾)))
43	42	ralunsn 4895	. . . . . . . . . . . 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 482	. . . . . . . . . . 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 482	. . . . . . . . . 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 712	. . . . . . . . 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 6838	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝑁–1-1-onto→𝑁 → dom 𝑔 = 𝑁)
48	47	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) → dom 𝑔 = 𝑁)
49	48	adantr 482	. . . . . . . . . . . 12 ⊢ (((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) ∧ (𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾)) → dom 𝑔 = 𝑁)
50		difsnid 4814	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ 𝑁 → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) = 𝑁)
51	50	eqcomd 2739	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ 𝑁 → 𝑁 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
52	49, 51	sylan9eqr 2795	. . . . . . . . . . 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 482	. . . . . . . . . 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 3326	. . . . . . . . 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 257	. . . . . . . 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 6836	. . . . . . . . . . . 12 ⊢ (𝑔:𝑁–1-1-onto→𝑁 → Fun 𝑔)
57	56	adantr 482	. . . . . . . . . . 11 ⊢ ((𝑔:𝑁–1-1-onto→𝑁 ∧ (𝑔‘𝐾) = 𝐾) → Fun 𝑔)
58		f1ofun 6836	. . . . . . . . . . . 12 ⊢ (𝑝:𝑁–1-1-onto→𝑁 → Fun 𝑝)
59	58	adantr 482	. . . . . . . . . . 11 ⊢ ((𝑝:𝑁–1-1-onto→𝑁 ∧ (𝑝‘𝐾) = 𝐾) → Fun 𝑝)
60	57, 59	anim12i 614	. . . . . . . . . 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 726	. . . . . . . . 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 7038	. . . . . . . . 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 712	. . . . . . 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 414	. . . . . 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 595	. . . 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 3201	. 2 ⊢ (𝐾 ∈ 𝑁 → ∀𝑔 ∈ 𝑄 ∀𝑝 ∈ 𝑄 ((𝐻‘𝑔) = (𝐻‘𝑝) → 𝑔 = 𝑝))
70		dff13 7254	. 2 ⊢ (𝐻:𝑄–1-1→𝑆 ↔ (𝐻:𝑄⟶𝑆 ∧ ∀𝑔 ∈ 𝑄 ∀𝑝 ∈ 𝑄 ((𝐻‘𝑔) = (𝐻‘𝑝) → 𝑔 = 𝑝)))
71	5, 69, 70	sylanbrc 584	1 ⊢ (𝐾 ∈ 𝑁 → 𝐻:𝑄–1-1→𝑆)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  {crab 3433  Vcvv 3475   ∖ cdif 3946   ∪ cun 3947   ⊆ wss 3949  {csn 4629   ↦ cmpt 5232  dom cdm 5677   ↾ cres 5679  Fun wfun 6538   Fn wfn 6539  ⟶wf 6540  –1-1→wf1 6541  –1-1-onto→wf1o 6543  ‘cfv 6544  Basecbs 17144  SymGrpcsymg 19234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-tset 17216  df-efmnd 18750  df-symg 19235

This theorem is referenced by:  symgfixf1o  19308