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Theorem efgredeu 19356

Description: There is a unique reduced word equivalent to a given word. (Contributed by Mario Carneiro, 1-Oct-2015.)

**Hypotheses**
Ref	Expression
efgval.w	⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o))
efgval.r	⊢ ∼ = ( ~_FG ‘𝐼)
efgval2.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)
efgval2.t	⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
efgred.d	⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
efgred.s	⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))

**Assertion**
Ref	Expression
efgredeu	⊢ (𝐴 ∈ 𝑊 → ∃!𝑑 ∈ 𝐷 𝑑 ∼ 𝐴)

Distinct variable groups: 𝐴,𝑑 𝑦,𝑧 𝑡,𝑛,𝑣,𝑤,𝑦,𝑧,𝑚,𝑥 𝑚,𝑀 𝑥,𝑛,𝑀,𝑡,𝑣,𝑤 𝑘,𝑚,𝑡,𝑥,𝑇 𝑘,𝑑,𝑚,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧,𝑊 ∼ ,𝑑,𝑚,𝑡,𝑥,𝑦,𝑧 𝑆,𝑑 𝑚,𝐼,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧 𝐷,𝑑,𝑚,𝑡 Allowed substitution hints: 𝐴(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛) 𝐷(𝑥,𝑦,𝑧,𝑤,𝑣,𝑘,𝑛) ∼ (𝑤,𝑣,𝑘,𝑛) 𝑆(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛) 𝑇(𝑦,𝑧,𝑤,𝑣,𝑛,𝑑) 𝐼(𝑘,𝑑) 𝑀(𝑦,𝑧,𝑘,𝑑)

Proof of Theorem efgredeu Dummy variables 𝑎 𝑏 𝑐 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgval.w	. . . . 5 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o))
2		efgval.r	. . . . 5 ⊢ ∼ = ( ~_FG ‘𝐼)
3		efgval2.m	. . . . 5 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)
4		efgval2.t	. . . . 5 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
5		efgred.d	. . . . 5 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
6		efgred.s	. . . . 5 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))
7	1, 2, 3, 4, 5, 6	efgsfo 19343	. . . 4 ⊢ 𝑆:dom 𝑆–onto→𝑊
8		foelrn 6979	. . . 4 ⊢ ((𝑆:dom 𝑆–onto→𝑊 ∧ 𝐴 ∈ 𝑊) → ∃𝑎 ∈ dom 𝑆 𝐴 = (𝑆‘𝑎))
9	7, 8	mpan 687	. . 3 ⊢ (𝐴 ∈ 𝑊 → ∃𝑎 ∈ dom 𝑆 𝐴 = (𝑆‘𝑎))
10	1, 2, 3, 4, 5, 6	efgsdm 19334	. . . . . . 7 ⊢ (𝑎 ∈ dom 𝑆 ↔ (𝑎 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑎‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝑎))(𝑎‘𝑖) ∈ ran (𝑇‘(𝑎‘(𝑖 − 1)))))
11	10	simp2bi 1145	. . . . . 6 ⊢ (𝑎 ∈ dom 𝑆 → (𝑎‘0) ∈ 𝐷)
12	1, 2, 3, 4, 5, 6	efgsrel 19338	. . . . . . 7 ⊢ (𝑎 ∈ dom 𝑆 → (𝑎‘0) ∼ (𝑆‘𝑎))
13	12	adantl 482	. . . . . 6 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑎 ∈ dom 𝑆) → (𝑎‘0) ∼ (𝑆‘𝑎))
14		breq1 5082	. . . . . . 7 ⊢ (𝑑 = (𝑎‘0) → (𝑑 ∼ (𝑆‘𝑎) ↔ (𝑎‘0) ∼ (𝑆‘𝑎)))
15	14	rspcev 3561	. . . . . 6 ⊢ (((𝑎‘0) ∈ 𝐷 ∧ (𝑎‘0) ∼ (𝑆‘𝑎)) → ∃𝑑 ∈ 𝐷 𝑑 ∼ (𝑆‘𝑎))
16	11, 13, 15	syl2an2 683	. . . . 5 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑎 ∈ dom 𝑆) → ∃𝑑 ∈ 𝐷 𝑑 ∼ (𝑆‘𝑎))
17		breq2 5083	. . . . . 6 ⊢ (𝐴 = (𝑆‘𝑎) → (𝑑 ∼ 𝐴 ↔ 𝑑 ∼ (𝑆‘𝑎)))
18	17	rexbidv 3228	. . . . 5 ⊢ (𝐴 = (𝑆‘𝑎) → (∃𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ↔ ∃𝑑 ∈ 𝐷 𝑑 ∼ (𝑆‘𝑎)))
19	16, 18	syl5ibrcom 246	. . . 4 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑎 ∈ dom 𝑆) → (𝐴 = (𝑆‘𝑎) → ∃𝑑 ∈ 𝐷 𝑑 ∼ 𝐴))
20	19	rexlimdva 3215	. . 3 ⊢ (𝐴 ∈ 𝑊 → (∃𝑎 ∈ dom 𝑆 𝐴 = (𝑆‘𝑎) → ∃𝑑 ∈ 𝐷 𝑑 ∼ 𝐴))
21	9, 20	mpd 15	. 2 ⊢ (𝐴 ∈ 𝑊 → ∃𝑑 ∈ 𝐷 𝑑 ∼ 𝐴)
22	1, 2	efger 19322	. . . . . . 7 ⊢ ∼ Er 𝑊
23	22	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → ∼ Er 𝑊)
24		simprl 768	. . . . . 6 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → 𝑑 ∼ 𝐴)
25		simprr 770	. . . . . 6 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → 𝑐 ∼ 𝐴)
26	23, 24, 25	ertr4d 8500	. . . . 5 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → 𝑑 ∼ 𝑐)
27	1, 2, 3, 4, 5, 6	efgrelex 19355	. . . . . 6 ⊢ (𝑑 ∼ 𝑐 → ∃𝑎 ∈ (^◡𝑆 “ {𝑑})∃𝑏 ∈ (^◡𝑆 “ {𝑐})(𝑎‘0) = (𝑏‘0))
28		fofn 6688	. . . . . . . . . . . . . 14 ⊢ (𝑆:dom 𝑆–onto→𝑊 → 𝑆 Fn dom 𝑆)
29		fniniseg 6934	. . . . . . . . . . . . . 14 ⊢ (𝑆 Fn dom 𝑆 → (𝑎 ∈ (^◡𝑆 “ {𝑑}) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆‘𝑎) = 𝑑)))
30	7, 28, 29	mp2b 10	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ↔ (𝑎 ∈ dom 𝑆 ∧ (𝑆‘𝑎) = 𝑑))
31	30	simplbi 498	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (^◡𝑆 “ {𝑑}) → 𝑎 ∈ dom 𝑆)
32	31	ad2antrl 725	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → 𝑎 ∈ dom 𝑆)
33	1, 2, 3, 4, 5, 6	efgsval 19335	. . . . . . . . . . 11 ⊢ (𝑎 ∈ dom 𝑆 → (𝑆‘𝑎) = (𝑎‘((♯‘𝑎) − 1)))
34	32, 33	syl 17	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → (𝑆‘𝑎) = (𝑎‘((♯‘𝑎) − 1)))
35	30	simprbi 497	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (^◡𝑆 “ {𝑑}) → (𝑆‘𝑎) = 𝑑)
36	35	ad2antrl 725	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → (𝑆‘𝑎) = 𝑑)
37		simpllr 773	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷))
38	37	simpld 495	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → 𝑑 ∈ 𝐷)
39	36, 38	eqeltrd 2841	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → (𝑆‘𝑎) ∈ 𝐷)
40	1, 2, 3, 4, 5, 6	efgs1b 19340	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ dom 𝑆 → ((𝑆‘𝑎) ∈ 𝐷 ↔ (♯‘𝑎) = 1))
41	32, 40	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → ((𝑆‘𝑎) ∈ 𝐷 ↔ (♯‘𝑎) = 1))
42	39, 41	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → (♯‘𝑎) = 1)
43	42	oveq1d 7286	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → ((♯‘𝑎) − 1) = (1 − 1))
44		1m1e0 12045	. . . . . . . . . . . 12 ⊢ (1 − 1) = 0
45	43, 44	eqtrdi 2796	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → ((♯‘𝑎) − 1) = 0)
46	45	fveq2d 6775	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → (𝑎‘((♯‘𝑎) − 1)) = (𝑎‘0))
47	34, 36, 46	3eqtr3rd 2789	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → (𝑎‘0) = 𝑑)
48		fniniseg 6934	. . . . . . . . . . . . . 14 ⊢ (𝑆 Fn dom 𝑆 → (𝑏 ∈ (^◡𝑆 “ {𝑐}) ↔ (𝑏 ∈ dom 𝑆 ∧ (𝑆‘𝑏) = 𝑐)))
49	7, 28, 48	mp2b 10	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (^◡𝑆 “ {𝑐}) ↔ (𝑏 ∈ dom 𝑆 ∧ (𝑆‘𝑏) = 𝑐))
50	49	simplbi 498	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ (^◡𝑆 “ {𝑐}) → 𝑏 ∈ dom 𝑆)
51	50	ad2antll 726	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → 𝑏 ∈ dom 𝑆)
52	1, 2, 3, 4, 5, 6	efgsval 19335	. . . . . . . . . . 11 ⊢ (𝑏 ∈ dom 𝑆 → (𝑆‘𝑏) = (𝑏‘((♯‘𝑏) − 1)))
53	51, 52	syl 17	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → (𝑆‘𝑏) = (𝑏‘((♯‘𝑏) − 1)))
54	49	simprbi 497	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (^◡𝑆 “ {𝑐}) → (𝑆‘𝑏) = 𝑐)
55	54	ad2antll 726	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → (𝑆‘𝑏) = 𝑐)
56	37	simprd 496	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → 𝑐 ∈ 𝐷)
57	55, 56	eqeltrd 2841	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → (𝑆‘𝑏) ∈ 𝐷)
58	1, 2, 3, 4, 5, 6	efgs1b 19340	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ dom 𝑆 → ((𝑆‘𝑏) ∈ 𝐷 ↔ (♯‘𝑏) = 1))
59	51, 58	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → ((𝑆‘𝑏) ∈ 𝐷 ↔ (♯‘𝑏) = 1))
60	57, 59	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → (♯‘𝑏) = 1)
61	60	oveq1d 7286	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → ((♯‘𝑏) − 1) = (1 − 1))
62	61, 44	eqtrdi 2796	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → ((♯‘𝑏) − 1) = 0)
63	62	fveq2d 6775	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → (𝑏‘((♯‘𝑏) − 1)) = (𝑏‘0))
64	53, 55, 63	3eqtr3rd 2789	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → (𝑏‘0) = 𝑐)
65	47, 64	eqeq12d 2756	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → ((𝑎‘0) = (𝑏‘0) ↔ 𝑑 = 𝑐))
66	65	biimpd 228	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → ((𝑎‘0) = (𝑏‘0) → 𝑑 = 𝑐))
67	66	rexlimdvva 3225	. . . . . 6 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → (∃𝑎 ∈ (^◡𝑆 “ {𝑑})∃𝑏 ∈ (^◡𝑆 “ {𝑐})(𝑎‘0) = (𝑏‘0) → 𝑑 = 𝑐))
68	27, 67	syl5 34	. . . . 5 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → (𝑑 ∼ 𝑐 → 𝑑 = 𝑐))
69	26, 68	mpd 15	. . . 4 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → 𝑑 = 𝑐)
70	69	ex 413	. . 3 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) → ((𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴) → 𝑑 = 𝑐))
71	70	ralrimivva 3117	. 2 ⊢ (𝐴 ∈ 𝑊 → ∀𝑑 ∈ 𝐷 ∀𝑐 ∈ 𝐷 ((𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴) → 𝑑 = 𝑐))
72		breq1 5082	. . 3 ⊢ (𝑑 = 𝑐 → (𝑑 ∼ 𝐴 ↔ 𝑐 ∼ 𝐴))
73	72	reu4 3670	. 2 ⊢ (∃!𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ↔ (∃𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ∧ ∀𝑑 ∈ 𝐷 ∀𝑐 ∈ 𝐷 ((𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴) → 𝑑 = 𝑐)))
74	21, 71, 73	sylanbrc 583	1 ⊢ (𝐴 ∈ 𝑊 → ∃!𝑑 ∈ 𝐷 𝑑 ∼ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∀wral 3066 ∃wrex 3067 ∃!wreu 3068 {crab 3070 ∖ cdif 3889 ∅c0 4262 {csn 4567 ⟨cop 4573 ⟨cotp 4575 ∪ ciun 4930 class class class wbr 5079 ↦ cmpt 5162 I cid 5489 × cxp 5588 ^◡ccnv 5589 dom cdm 5590 ran crn 5591 “ cima 5593 Fn wfn 6427 –onto→wfo 6430 ‘cfv 6432 (class class class)co 7271 ∈ cmpo 7273 1_oc1o 8281 2_oc2o 8282 Er wer 8478 0cc0 10872 1c1 10873 − cmin 11205 ...cfz 13238 ..^cfzo 13381 ♯chash 14042 Word cword 14215 splice csplice 14460 ⟨“cs2 14552 ~_FG cefg 19310

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-ot 4576 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-2o 8289 df-er 8481 df-ec 8483 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12582 df-rp 12730 df-fz 13239 df-fzo 13382 df-hash 14043 df-word 14216 df-concat 14272 df-s1 14299 df-substr 14352 df-pfx 14382 df-splice 14461 df-s2 14559 df-efg 19313

This theorem is referenced by: efgred2 19357 frgpnabllem2 19473

W3C validator