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Theorem efgredeu 19620

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 19607	. . . 4 ⊢ 𝑆:dom 𝑆–onto→𝑊
8		foelrn 7108	. . . 4 ⊢ ((𝑆:dom 𝑆–onto→𝑊 ∧ 𝐴 ∈ 𝑊) → ∃𝑎 ∈ dom 𝑆 𝐴 = (𝑆‘𝑎))
9	7, 8	mpan 689	. . 3 ⊢ (𝐴 ∈ 𝑊 → ∃𝑎 ∈ dom 𝑆 𝐴 = (𝑆‘𝑎))
10	1, 2, 3, 4, 5, 6	efgsdm 19598	. . . . . . 7 ⊢ (𝑎 ∈ dom 𝑆 ↔ (𝑎 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑎‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝑎))(𝑎‘𝑖) ∈ ran (𝑇‘(𝑎‘(𝑖 − 1)))))
11	10	simp2bi 1147	. . . . . 6 ⊢ (𝑎 ∈ dom 𝑆 → (𝑎‘0) ∈ 𝐷)
12	1, 2, 3, 4, 5, 6	efgsrel 19602	. . . . . . 7 ⊢ (𝑎 ∈ dom 𝑆 → (𝑎‘0) ∼ (𝑆‘𝑎))
13	12	adantl 483	. . . . . 6 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑎 ∈ dom 𝑆) → (𝑎‘0) ∼ (𝑆‘𝑎))
14		breq1 5152	. . . . . . 7 ⊢ (𝑑 = (𝑎‘0) → (𝑑 ∼ (𝑆‘𝑎) ↔ (𝑎‘0) ∼ (𝑆‘𝑎)))
15	14	rspcev 3613	. . . . . 6 ⊢ (((𝑎‘0) ∈ 𝐷 ∧ (𝑎‘0) ∼ (𝑆‘𝑎)) → ∃𝑑 ∈ 𝐷 𝑑 ∼ (𝑆‘𝑎))
16	11, 13, 15	syl2an2 685	. . . . 5 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑎 ∈ dom 𝑆) → ∃𝑑 ∈ 𝐷 𝑑 ∼ (𝑆‘𝑎))
17		breq2 5153	. . . . . 6 ⊢ (𝐴 = (𝑆‘𝑎) → (𝑑 ∼ 𝐴 ↔ 𝑑 ∼ (𝑆‘𝑎)))
18	17	rexbidv 3179	. . . . 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 3156	. . 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 19586	. . . . . . 7 ⊢ ∼ Er 𝑊
23	22	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → ∼ Er 𝑊)
24		simprl 770	. . . . . 6 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → 𝑑 ∼ 𝐴)
25		simprr 772	. . . . . 6 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → 𝑐 ∼ 𝐴)
26	23, 24, 25	ertr4d 8722	. . . . 5 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → 𝑑 ∼ 𝑐)
27	1, 2, 3, 4, 5, 6	efgrelex 19619	. . . . . 6 ⊢ (𝑑 ∼ 𝑐 → ∃𝑎 ∈ (^◡𝑆 “ {𝑑})∃𝑏 ∈ (^◡𝑆 “ {𝑐})(𝑎‘0) = (𝑏‘0))
28		fofn 6808	. . . . . . . . . . . . . 14 ⊢ (𝑆:dom 𝑆–onto→𝑊 → 𝑆 Fn dom 𝑆)
29		fniniseg 7062	. . . . . . . . . . . . . 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 499	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (^◡𝑆 “ {𝑑}) → 𝑎 ∈ dom 𝑆)
32	31	ad2antrl 727	. . . . . . . . . . 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 19599	. . . . . . . . . . 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 498	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (^◡𝑆 “ {𝑑}) → (𝑆‘𝑎) = 𝑑)
36	35	ad2antrl 727	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → (𝑆‘𝑎) = 𝑑)
37		simpllr 775	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷))
38	37	simpld 496	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → 𝑑 ∈ 𝐷)
39	36, 38	eqeltrd 2834	. . . . . . . . . . . . . 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 19604	. . . . . . . . . . . . . . 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 7424	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → ((♯‘𝑎) − 1) = (1 − 1))
44		1m1e0 12284	. . . . . . . . . . . 12 ⊢ (1 − 1) = 0
45	43, 44	eqtrdi 2789	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → ((♯‘𝑎) − 1) = 0)
46	45	fveq2d 6896	. . . . . . . . . 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 2782	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → (𝑎‘0) = 𝑑)
48		fniniseg 7062	. . . . . . . . . . . . . 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 499	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ (^◡𝑆 “ {𝑐}) → 𝑏 ∈ dom 𝑆)
51	50	ad2antll 728	. . . . . . . . . . 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 19599	. . . . . . . . . . 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 498	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (^◡𝑆 “ {𝑐}) → (𝑆‘𝑏) = 𝑐)
55	54	ad2antll 728	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → (𝑆‘𝑏) = 𝑐)
56	37	simprd 497	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → 𝑐 ∈ 𝐷)
57	55, 56	eqeltrd 2834	. . . . . . . . . . . . . 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 19604	. . . . . . . . . . . . . . 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 7424	. . . . . . . . . . . 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 2789	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → ((♯‘𝑏) − 1) = 0)
63	62	fveq2d 6896	. . . . . . . . . 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 2782	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (^◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (^◡𝑆 “ {𝑐}))) → (𝑏‘0) = 𝑐)
65	47, 64	eqeq12d 2749	. . . . . . . 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 3212	. . . . . 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 414	. . 3 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) → ((𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴) → 𝑑 = 𝑐))
71	70	ralrimivva 3201	. 2 ⊢ (𝐴 ∈ 𝑊 → ∀𝑑 ∈ 𝐷 ∀𝑐 ∈ 𝐷 ((𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴) → 𝑑 = 𝑐))
72		breq1 5152	. . 3 ⊢ (𝑑 = 𝑐 → (𝑑 ∼ 𝐴 ↔ 𝑐 ∼ 𝐴))
73	72	reu4 3728	. 2 ⊢ (∃!𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ↔ (∃𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ∧ ∀𝑑 ∈ 𝐷 ∀𝑐 ∈ 𝐷 ((𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴) → 𝑑 = 𝑐)))
74	21, 71, 73	sylanbrc 584	1 ⊢ (𝐴 ∈ 𝑊 → ∃!𝑑 ∈ 𝐷 𝑑 ∼ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ∃!wreu 3375 {crab 3433 ∖ cdif 3946 ∅c0 4323 {csn 4629 ⟨cop 4635 ⟨cotp 4637 ∪ ciun 4998 class class class wbr 5149 ↦ cmpt 5232 I cid 5574 × cxp 5675 ^◡ccnv 5676 dom cdm 5677 ran crn 5678 “ cima 5680 Fn wfn 6539 –onto→wfo 6542 ‘cfv 6544 (class class class)co 7409 ∈ cmpo 7411 1_oc1o 8459 2_oc2o 8460 Er wer 8700 0cc0 11110 1c1 11111 − cmin 11444 ...cfz 13484 ..^cfzo 13627 ♯chash 14290 Word cword 14464 splice csplice 14699 ⟨“cs2 14792 ~_FG cefg 19574

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-rep 5286 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-rmo 3377 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-op 4636 df-ot 4638 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 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-2o 8467 df-er 8703 df-ec 8705 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9934 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-n0 12473 df-xnn0 12545 df-z 12559 df-uz 12823 df-rp 12975 df-fz 13485 df-fzo 13628 df-hash 14291 df-word 14465 df-concat 14521 df-s1 14546 df-substr 14591 df-pfx 14621 df-splice 14700 df-s2 14799 df-efg 19577

This theorem is referenced by: efgred2 19621 frgpnabllem2 19742

W3C validator