Theorem efgredeu 19959

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 × 2o))
efgval.r	⊢ ∼ = ( ~FG ‘𝐼)
efgval2.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
efgval2.t	⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (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 × 2o))
2		efgval.r	. . . . 5 ⊢ ∼ = ( ~FG ‘𝐼)
3		efgval2.m	. . . . 5 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
4		efgval2.t	. . . . 5 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (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 19946	. . . 4 ⊢ 𝑆:dom 𝑆–onto→𝑊
8		foelrn 7105	. . . 4 ⊢ ((𝑆:dom 𝑆–onto→𝑊 ∧ 𝐴 ∈ 𝑊) → ∃𝑎 ∈ dom 𝑆 𝐴 = (𝑆‘𝑎))
9	7, 8	mpan 703	. . 3 ⊢ (𝐴 ∈ 𝑊 → ∃𝑎 ∈ dom 𝑆 𝐴 = (𝑆‘𝑎))
10	1, 2, 3, 4, 5, 6	efgsdm 19937	. . . . . . 7 ⊢ (𝑎 ∈ dom 𝑆 ↔ (𝑎 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝑎‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝑎))(𝑎‘𝑖) ∈ ran (𝑇‘(𝑎‘(𝑖 − 1)))))
11	10	simp2bi 1164	. . . . . 6 ⊢ (𝑎 ∈ dom 𝑆 → (𝑎‘0) ∈ 𝐷)
12	1, 2, 3, 4, 5, 6	efgsrel 19941	. . . . . . 7 ⊢ (𝑎 ∈ dom 𝑆 → (𝑎‘0) ∼ (𝑆‘𝑎))
13	12	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑎 ∈ dom 𝑆) → (𝑎‘0) ∼ (𝑆‘𝑎))
14		breq1 5106	. . . . . . 7 ⊢ (𝑑 = (𝑎‘0) → (𝑑 ∼ (𝑆‘𝑎) ↔ (𝑎‘0) ∼ (𝑆‘𝑎)))
15	14	rspcev 3577	. . . . . 6 ⊢ (((𝑎‘0) ∈ 𝐷 ∧ (𝑎‘0) ∼ (𝑆‘𝑎)) → ∃𝑑 ∈ 𝐷 𝑑 ∼ (𝑆‘𝑎))
16	11, 13, 15	syl2an2 699	. . . . 5 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑎 ∈ dom 𝑆) → ∃𝑑 ∈ 𝐷 𝑑 ∼ (𝑆‘𝑎))
17		breq2 5107	. . . . . 6 ⊢ (𝐴 = (𝑆‘𝑎) → (𝑑 ∼ 𝐴 ↔ 𝑑 ∼ (𝑆‘𝑎)))
18	17	rexbidv 3187	. . . . 5 ⊢ (𝐴 = (𝑆‘𝑎) → (∃𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ↔ ∃𝑑 ∈ 𝐷 𝑑 ∼ (𝑆‘𝑎)))
19	16, 18	syl5ibrcom 250	. . . 4 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑎 ∈ dom 𝑆) → (𝐴 = (𝑆‘𝑎) → ∃𝑑 ∈ 𝐷 𝑑 ∼ 𝐴))
20	19	rexlimdva 3164	. . 3 ⊢ (𝐴 ∈ 𝑊 → (∃𝑎 ∈ dom 𝑆 𝐴 = (𝑆‘𝑎) → ∃𝑑 ∈ 𝐷 𝑑 ∼ 𝐴))
21	9, 20	mpd 16	. 2 ⊢ (𝐴 ∈ 𝑊 → ∃𝑑 ∈ 𝐷 𝑑 ∼ 𝐴)
22	1, 2	efger 19925	. . . . . . 7 ⊢ ∼ Er 𝑊
23	22	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → ∼ Er 𝑊)
24		simprl 783	. . . . . 6 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → 𝑑 ∼ 𝐴)
25		simprr 785	. . . . . 6 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → 𝑐 ∼ 𝐴)
26	23, 24, 25	ertr4d 8730	. . . . 5 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → 𝑑 ∼ 𝑐)
27	1, 2, 3, 4, 5, 6	efgrelex 19958	. . . . . 6 ⊢ (𝑑 ∼ 𝑐 → ∃𝑎 ∈ (◡𝑆 “ {𝑑})∃𝑏 ∈ (◡𝑆 “ {𝑐})(𝑎‘0) = (𝑏‘0))
28		fofn 6796	. . . . . . . . . . . . . 14 ⊢ (𝑆:dom 𝑆–onto→𝑊 → 𝑆 Fn dom 𝑆)
29		fniniseg 7057	. . . . . . . . . . . . . 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 502	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (◡𝑆 “ {𝑑}) → 𝑎 ∈ dom 𝑆)
32	31	ad2antrl 741	. . . . . . . . . . 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 19938	. . . . . . . . . . 11 ⊢ (𝑎 ∈ dom 𝑆 → (𝑆‘𝑎) = (𝑎‘((♯‘𝑎) − 1)))
34	32, 33	syl 18	. . . . . . . . . 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 503	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (◡𝑆 “ {𝑑}) → (𝑆‘𝑎) = 𝑑)
36	35	ad2antrl 741	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → (𝑆‘𝑎) = 𝑑)
37		simpllr 788	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷))
38	37	simpld 500	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → 𝑑 ∈ 𝐷)
39	36, 38	eqeltrd 2861	. . . . . . . . . . . . . 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 19943	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ dom 𝑆 → ((𝑆‘𝑎) ∈ 𝐷 ↔ (♯‘𝑎) = 1))
41	32, 40	syl 18	. . . . . . . . . . . . . 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 235	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → (♯‘𝑎) = 1)
43	42	oveq1d 7433	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → ((♯‘𝑎) − 1) = (1 − 1))
44		1m1e0 12408	. . . . . . . . . . . 12 ⊢ (1 − 1) = 0
45	43, 44	eqtrdi 2812	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → ((♯‘𝑎) − 1) = 0)
46	45	fveq2d 6887	. . . . . . . . . 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 2805	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → (𝑎‘0) = 𝑑)
48		fniniseg 7057	. . . . . . . . . . . . . 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 502	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ (◡𝑆 “ {𝑐}) → 𝑏 ∈ dom 𝑆)
51	50	ad2antll 742	. . . . . . . . . . 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 19938	. . . . . . . . . . 11 ⊢ (𝑏 ∈ dom 𝑆 → (𝑆‘𝑏) = (𝑏‘((♯‘𝑏) − 1)))
53	51, 52	syl 18	. . . . . . . . . 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 503	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (◡𝑆 “ {𝑐}) → (𝑆‘𝑏) = 𝑐)
55	54	ad2antll 742	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → (𝑆‘𝑏) = 𝑐)
56	37	simprd 501	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → 𝑐 ∈ 𝐷)
57	55, 56	eqeltrd 2861	. . . . . . . . . . . . . 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 19943	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ dom 𝑆 → ((𝑆‘𝑏) ∈ 𝐷 ↔ (♯‘𝑏) = 1))
59	51, 58	syl 18	. . . . . . . . . . . . . 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 235	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → (♯‘𝑏) = 1)
61	60	oveq1d 7433	. . . . . . . . . . . 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 2812	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → ((♯‘𝑏) − 1) = 0)
63	62	fveq2d 6887	. . . . . . . . . 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 2805	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) ∧ (𝑎 ∈ (◡𝑆 “ {𝑑}) ∧ 𝑏 ∈ (◡𝑆 “ {𝑐}))) → (𝑏‘0) = 𝑐)
65	47, 64	eqeq12d 2777	. . . . . . . 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 232	. . . . . . 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 3220	. . . . . 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 35	. . . . 5 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → (𝑑 ∼ 𝑐 → 𝑑 = 𝑐))
69	26, 68	mpd 16	. . . 4 ⊢ (((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) ∧ (𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴)) → 𝑑 = 𝑐)
70	69	ex 418	. . 3 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) → ((𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴) → 𝑑 = 𝑐))
71	70	ralrimivva 3206	. 2 ⊢ (𝐴 ∈ 𝑊 → ∀𝑑 ∈ 𝐷 ∀𝑐 ∈ 𝐷 ((𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴) → 𝑑 = 𝑐))
72		breq1 5106	. . 3 ⊢ (𝑑 = 𝑐 → (𝑑 ∼ 𝐴 ↔ 𝑐 ∼ 𝐴))
73	72	reu4 3689	. 2 ⊢ (∃!𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ↔ (∃𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ∧ ∀𝑑 ∈ 𝐷 ∀𝑐 ∈ 𝐷 ((𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴) → 𝑑 = 𝑐)))
74	21, 71, 73	sylanbrc 595	1 ⊢ (𝐴 ∈ 𝑊 → ∃!𝑑 ∈ 𝐷 𝑑 ∼ 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  ∃!wreu 3364  {crab 3413   ∖ cdif 3896  ∅c0 4279  {csn 4584  ⟨cop 4590  ⟨cotp 4592  ∪ ciun 4951   class class class wbr 5103   ↦ cmpt 5186   I cid 5545   × cxp 5649  ^◡ccnv 5650  dom cdm 5651  ran crn 5652   “ cima 5654   Fn wfn 6532  –onto→wfo 6535  ‘cfv 6537  (class class class)co 7418   ∈ cmpo 7420  1_oc1o 8462  2_oc2o 8463   Er wer 8707  0cc0 11193  1c1 11194   − cmin 11534  ...cfz 13632  ..^cfzo 13781  ♯chash 14467  Word cword 14651   splice csplice 14891  ⟨“cs2 14985   ~_FG cefg 19913

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-er 8710  df-ec 8712  df-map 8842  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-n0 12600  df-xnn0 12673  df-z 12687  df-uz 12959  df-rp 13114  df-fz 13633  df-fzo 13782  df-hash 14468  df-word 14652  df-concat 14709  df-s1 14736  df-substr 14782  df-pfx 14814  df-splice 14892  df-s2 14992  df-efg 19916

This theorem is used by:  efgred2  19960  frgpnabllem2  20081