Theorem riotaeqimp 7414

Description: If two restricted iota descriptors for an equality are equal, then the terms of the equality are equal. (Contributed by AV, 6-Dec-2020.)

Hypotheses

Ref	Expression
riotaeqimp.i	⊢ 𝐼 = (℩𝑎 ∈ 𝑉 𝑋 = 𝐴)
riotaeqimp.j	⊢ 𝐽 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴)
riotaeqimp.x	⊢ (𝜑 → ∃!𝑎 ∈ 𝑉 𝑋 = 𝐴)
riotaeqimp.y	⊢ (𝜑 → ∃!𝑎 ∈ 𝑉 𝑌 = 𝐴)

Assertion

Ref	Expression
riotaeqimp	⊢ ((𝜑 ∧ 𝐼 = 𝐽) → 𝑋 = 𝑌)

Distinct variable groups:   𝐼,𝑎   𝐽,𝑎   𝑉,𝑎   𝑋,𝑎   𝑌,𝑎

Allowed substitution hints:   𝜑(𝑎)   𝐴(𝑎)

Proof of Theorem riotaeqimp

Step	Hyp	Ref	Expression
1		riotaeqimp.j	. . . . . . 7 ⊢ 𝐽 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴)
2	1	eqcomi 2746	. . . . . 6 ⊢ (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) = 𝐽
3	2	eqeq2i 2750	. . . . 5 ⊢ (𝐼 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) ↔ 𝐼 = 𝐽)
4	3	a1i 11	. . . 4 ⊢ (𝜑 → (𝐼 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) ↔ 𝐼 = 𝐽))
5	4	bicomd 223	. . 3 ⊢ (𝜑 → (𝐼 = 𝐽 ↔ 𝐼 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴)))
6	5	biimpa 476	. 2 ⊢ ((𝜑 ∧ 𝐼 = 𝐽) → 𝐼 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴))
7		riotaeqimp.i	. . . . 5 ⊢ 𝐼 = (℩𝑎 ∈ 𝑉 𝑋 = 𝐴)
8	7	eqeq1i 2742	. . . 4 ⊢ (𝐼 = 𝐽 ↔ (℩𝑎 ∈ 𝑉 𝑋 = 𝐴) = 𝐽)
9		riotaeqimp.y	. . . . . . 7 ⊢ (𝜑 → ∃!𝑎 ∈ 𝑉 𝑌 = 𝐴)
10		riotacl 7405	. . . . . . 7 ⊢ (∃!𝑎 ∈ 𝑉 𝑌 = 𝐴 → (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) ∈ 𝑉)
11	9, 10	syl 17	. . . . . 6 ⊢ (𝜑 → (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) ∈ 𝑉)
12	1, 11	eqeltrid 2845	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ 𝑉)
13		riotaeqimp.x	. . . . 5 ⊢ (𝜑 → ∃!𝑎 ∈ 𝑉 𝑋 = 𝐴)
14		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑎 𝐽 ∈ 𝑉
15		nfcvd 2906	. . . . . . 7 ⊢ (𝐽 ∈ 𝑉 → Ⅎ𝑎𝐽)
16		nfcvd 2906	. . . . . . . 8 ⊢ (𝐽 ∈ 𝑉 → Ⅎ𝑎𝑋)
17	15	nfcsb1d 3921	. . . . . . . 8 ⊢ (𝐽 ∈ 𝑉 → Ⅎ𝑎⦋𝐽 / 𝑎⦌𝐴)
18	16, 17	nfeqd 2916	. . . . . . 7 ⊢ (𝐽 ∈ 𝑉 → Ⅎ𝑎 𝑋 = ⦋𝐽 / 𝑎⦌𝐴)
19		id 22	. . . . . . 7 ⊢ (𝐽 ∈ 𝑉 → 𝐽 ∈ 𝑉)
20		csbeq1a 3913	. . . . . . . . 9 ⊢ (𝑎 = 𝐽 → 𝐴 = ⦋𝐽 / 𝑎⦌𝐴)
21	20	eqeq2d 2748	. . . . . . . 8 ⊢ (𝑎 = 𝐽 → (𝑋 = 𝐴 ↔ 𝑋 = ⦋𝐽 / 𝑎⦌𝐴))
22	21	adantl 481	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝑎 = 𝐽) → (𝑋 = 𝐴 ↔ 𝑋 = ⦋𝐽 / 𝑎⦌𝐴))
23	14, 15, 18, 19, 22	riota2df 7411	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ ∃!𝑎 ∈ 𝑉 𝑋 = 𝐴) → (𝑋 = ⦋𝐽 / 𝑎⦌𝐴 ↔ (℩𝑎 ∈ 𝑉 𝑋 = 𝐴) = 𝐽))
24	23	bicomd 223	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ ∃!𝑎 ∈ 𝑉 𝑋 = 𝐴) → ((℩𝑎 ∈ 𝑉 𝑋 = 𝐴) = 𝐽 ↔ 𝑋 = ⦋𝐽 / 𝑎⦌𝐴))
25	12, 13, 24	syl2anc 584	. . . 4 ⊢ (𝜑 → ((℩𝑎 ∈ 𝑉 𝑋 = 𝐴) = 𝐽 ↔ 𝑋 = ⦋𝐽 / 𝑎⦌𝐴))
26	8, 25	bitrid 283	. . 3 ⊢ (𝜑 → (𝐼 = 𝐽 ↔ 𝑋 = ⦋𝐽 / 𝑎⦌𝐴))
27	26	biimpa 476	. 2 ⊢ ((𝜑 ∧ 𝐼 = 𝐽) → 𝑋 = ⦋𝐽 / 𝑎⦌𝐴)
28		riotacl 7405	. . . . . . . 8 ⊢ (∃!𝑎 ∈ 𝑉 𝑋 = 𝐴 → (℩𝑎 ∈ 𝑉 𝑋 = 𝐴) ∈ 𝑉)
29	13, 28	syl 17	. . . . . . 7 ⊢ (𝜑 → (℩𝑎 ∈ 𝑉 𝑋 = 𝐴) ∈ 𝑉)
30	7, 29	eqeltrid 2845	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
31		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑎 𝐼 ∈ 𝑉
32		nfcvd 2906	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → Ⅎ𝑎𝐼)
33		nfcvd 2906	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → Ⅎ𝑎𝑌)
34	32	nfcsb1d 3921	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → Ⅎ𝑎⦋𝐼 / 𝑎⦌𝐴)
35	33, 34	nfeqd 2916	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → Ⅎ𝑎 𝑌 = ⦋𝐼 / 𝑎⦌𝐴)
36		id 22	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ 𝑉)
37		csbeq1a 3913	. . . . . . . . 9 ⊢ (𝑎 = 𝐼 → 𝐴 = ⦋𝐼 / 𝑎⦌𝐴)
38	37	eqeq2d 2748	. . . . . . . 8 ⊢ (𝑎 = 𝐼 → (𝑌 = 𝐴 ↔ 𝑌 = ⦋𝐼 / 𝑎⦌𝐴))
39	38	adantl 481	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑎 = 𝐼) → (𝑌 = 𝐴 ↔ 𝑌 = ⦋𝐼 / 𝑎⦌𝐴))
40	31, 32, 35, 36, 39	riota2df 7411	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ ∃!𝑎 ∈ 𝑉 𝑌 = 𝐴) → (𝑌 = ⦋𝐼 / 𝑎⦌𝐴 ↔ (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) = 𝐼))
41	30, 9, 40	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑌 = ⦋𝐼 / 𝑎⦌𝐴 ↔ (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) = 𝐼))
42		eqcom 2744	. . . . 5 ⊢ ((℩𝑎 ∈ 𝑉 𝑌 = 𝐴) = 𝐼 ↔ 𝐼 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴))
43	41, 42	bitrdi 287	. . . 4 ⊢ (𝜑 → (𝑌 = ⦋𝐼 / 𝑎⦌𝐴 ↔ 𝐼 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴)))
44	43	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐼 = 𝐽) → (𝑌 = ⦋𝐼 / 𝑎⦌𝐴 ↔ 𝐼 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴)))
45		csbeq1 3902	. . . . . . 7 ⊢ (𝐽 = 𝐼 → ⦋𝐽 / 𝑎⦌𝐴 = ⦋𝐼 / 𝑎⦌𝐴)
46	45	eqcoms 2745	. . . . . 6 ⊢ (𝐼 = 𝐽 → ⦋𝐽 / 𝑎⦌𝐴 = ⦋𝐼 / 𝑎⦌𝐴)
47		eqeq12 2754	. . . . . . 7 ⊢ ((𝑋 = ⦋𝐽 / 𝑎⦌𝐴 ∧ 𝑌 = ⦋𝐼 / 𝑎⦌𝐴) → (𝑋 = 𝑌 ↔ ⦋𝐽 / 𝑎⦌𝐴 = ⦋𝐼 / 𝑎⦌𝐴))
48	47	ancoms 458	. . . . . 6 ⊢ ((𝑌 = ⦋𝐼 / 𝑎⦌𝐴 ∧ 𝑋 = ⦋𝐽 / 𝑎⦌𝐴) → (𝑋 = 𝑌 ↔ ⦋𝐽 / 𝑎⦌𝐴 = ⦋𝐼 / 𝑎⦌𝐴))
49	46, 48	syl5ibrcom 247	. . . . 5 ⊢ (𝐼 = 𝐽 → ((𝑌 = ⦋𝐼 / 𝑎⦌𝐴 ∧ 𝑋 = ⦋𝐽 / 𝑎⦌𝐴) → 𝑋 = 𝑌))
50	49	expd 415	. . . 4 ⊢ (𝐼 = 𝐽 → (𝑌 = ⦋𝐼 / 𝑎⦌𝐴 → (𝑋 = ⦋𝐽 / 𝑎⦌𝐴 → 𝑋 = 𝑌)))
51	50	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝐼 = 𝐽) → (𝑌 = ⦋𝐼 / 𝑎⦌𝐴 → (𝑋 = ⦋𝐽 / 𝑎⦌𝐴 → 𝑋 = 𝑌)))
52	44, 51	sylbird 260	. 2 ⊢ ((𝜑 ∧ 𝐼 = 𝐽) → (𝐼 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) → (𝑋 = ⦋𝐽 / 𝑎⦌𝐴 → 𝑋 = 𝑌)))
53	6, 27, 52	mp2d 49	1 ⊢ ((𝜑 ∧ 𝐼 = 𝐽) → 𝑋 = 𝑌)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃!wreu 3378  ⦋csb 3899  ℩crio 7387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-un 3956  df-ss 3968  df-sn 4627  df-pr 4629  df-uni 4908  df-iota 6514  df-riota 7388

This theorem is referenced by:  uspgredg2v  29241