Theorem riotaeqimp 5952

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 2213	. . . . . 6 ⊢ (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) = 𝐽
3	2	eqeq2i 2220	. . . . 5 ⊢ (𝐼 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) ↔ 𝐼 = 𝐽)
4	3	a1i 9	. . . 4 ⊢ (𝜑 → (𝐼 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) ↔ 𝐼 = 𝐽))
5	4	bicomd 141	. . 3 ⊢ (𝜑 → (𝐼 = 𝐽 ↔ 𝐼 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴)))
6	5	biimpa 296	. 2 ⊢ ((𝜑 ∧ 𝐼 = 𝐽) → 𝐼 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴))
7		riotaeqimp.i	. . . . 5 ⊢ 𝐼 = (℩𝑎 ∈ 𝑉 𝑋 = 𝐴)
8	7	eqeq1i 2217	. . . 4 ⊢ (𝐼 = 𝐽 ↔ (℩𝑎 ∈ 𝑉 𝑋 = 𝐴) = 𝐽)
9		riotaeqimp.y	. . . . . . 7 ⊢ (𝜑 → ∃!𝑎 ∈ 𝑉 𝑌 = 𝐴)
10		riotacl 5943	. . . . . . 7 ⊢ (∃!𝑎 ∈ 𝑉 𝑌 = 𝐴 → (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) ∈ 𝑉)
11	9, 10	syl 14	. . . . . 6 ⊢ (𝜑 → (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) ∈ 𝑉)
12	1, 11	eqeltrid 2296	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ 𝑉)
13		riotaeqimp.x	. . . . 5 ⊢ (𝜑 → ∃!𝑎 ∈ 𝑉 𝑋 = 𝐴)
14		nfv 1554	. . . . . . 7 ⊢ Ⅎ𝑎 𝐽 ∈ 𝑉
15		nfcvd 2353	. . . . . . 7 ⊢ (𝐽 ∈ 𝑉 → Ⅎ𝑎𝐽)
16		nfcvd 2353	. . . . . . . 8 ⊢ (𝐽 ∈ 𝑉 → Ⅎ𝑎𝑋)
17	15	nfcsb1d 3135	. . . . . . . 8 ⊢ (𝐽 ∈ 𝑉 → Ⅎ𝑎⦋𝐽 / 𝑎⦌𝐴)
18	16, 17	nfeqd 2367	. . . . . . 7 ⊢ (𝐽 ∈ 𝑉 → Ⅎ𝑎 𝑋 = ⦋𝐽 / 𝑎⦌𝐴)
19		id 19	. . . . . . 7 ⊢ (𝐽 ∈ 𝑉 → 𝐽 ∈ 𝑉)
20		csbeq1a 3113	. . . . . . . . 9 ⊢ (𝑎 = 𝐽 → 𝐴 = ⦋𝐽 / 𝑎⦌𝐴)
21	20	eqeq2d 2221	. . . . . . . 8 ⊢ (𝑎 = 𝐽 → (𝑋 = 𝐴 ↔ 𝑋 = ⦋𝐽 / 𝑎⦌𝐴))
22	21	adantl 277	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝑎 = 𝐽) → (𝑋 = 𝐴 ↔ 𝑋 = ⦋𝐽 / 𝑎⦌𝐴))
23	14, 15, 18, 19, 22	riota2df 5949	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ ∃!𝑎 ∈ 𝑉 𝑋 = 𝐴) → (𝑋 = ⦋𝐽 / 𝑎⦌𝐴 ↔ (℩𝑎 ∈ 𝑉 𝑋 = 𝐴) = 𝐽))
24	23	bicomd 141	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ ∃!𝑎 ∈ 𝑉 𝑋 = 𝐴) → ((℩𝑎 ∈ 𝑉 𝑋 = 𝐴) = 𝐽 ↔ 𝑋 = ⦋𝐽 / 𝑎⦌𝐴))
25	12, 13, 24	syl2anc 411	. . . 4 ⊢ (𝜑 → ((℩𝑎 ∈ 𝑉 𝑋 = 𝐴) = 𝐽 ↔ 𝑋 = ⦋𝐽 / 𝑎⦌𝐴))
26	8, 25	bitrid 192	. . 3 ⊢ (𝜑 → (𝐼 = 𝐽 ↔ 𝑋 = ⦋𝐽 / 𝑎⦌𝐴))
27	26	biimpa 296	. 2 ⊢ ((𝜑 ∧ 𝐼 = 𝐽) → 𝑋 = ⦋𝐽 / 𝑎⦌𝐴)
28		riotacl 5943	. . . . . . . 8 ⊢ (∃!𝑎 ∈ 𝑉 𝑋 = 𝐴 → (℩𝑎 ∈ 𝑉 𝑋 = 𝐴) ∈ 𝑉)
29	13, 28	syl 14	. . . . . . 7 ⊢ (𝜑 → (℩𝑎 ∈ 𝑉 𝑋 = 𝐴) ∈ 𝑉)
30	7, 29	eqeltrid 2296	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
31		nfv 1554	. . . . . . 7 ⊢ Ⅎ𝑎 𝐼 ∈ 𝑉
32		nfcvd 2353	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → Ⅎ𝑎𝐼)
33		nfcvd 2353	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → Ⅎ𝑎𝑌)
34	32	nfcsb1d 3135	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → Ⅎ𝑎⦋𝐼 / 𝑎⦌𝐴)
35	33, 34	nfeqd 2367	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → Ⅎ𝑎 𝑌 = ⦋𝐼 / 𝑎⦌𝐴)
36		id 19	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ 𝑉)
37		csbeq1a 3113	. . . . . . . . 9 ⊢ (𝑎 = 𝐼 → 𝐴 = ⦋𝐼 / 𝑎⦌𝐴)
38	37	eqeq2d 2221	. . . . . . . 8 ⊢ (𝑎 = 𝐼 → (𝑌 = 𝐴 ↔ 𝑌 = ⦋𝐼 / 𝑎⦌𝐴))
39	38	adantl 277	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑎 = 𝐼) → (𝑌 = 𝐴 ↔ 𝑌 = ⦋𝐼 / 𝑎⦌𝐴))
40	31, 32, 35, 36, 39	riota2df 5949	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ ∃!𝑎 ∈ 𝑉 𝑌 = 𝐴) → (𝑌 = ⦋𝐼 / 𝑎⦌𝐴 ↔ (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) = 𝐼))
41	30, 9, 40	syl2anc 411	. . . . 5 ⊢ (𝜑 → (𝑌 = ⦋𝐼 / 𝑎⦌𝐴 ↔ (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) = 𝐼))
42		eqcom 2211	. . . . 5 ⊢ ((℩𝑎 ∈ 𝑉 𝑌 = 𝐴) = 𝐼 ↔ 𝐼 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴))
43	41, 42	bitrdi 196	. . . 4 ⊢ (𝜑 → (𝑌 = ⦋𝐼 / 𝑎⦌𝐴 ↔ 𝐼 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴)))
44	43	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝐼 = 𝐽) → (𝑌 = ⦋𝐼 / 𝑎⦌𝐴 ↔ 𝐼 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴)))
45		csbeq1 3107	. . . . . . 7 ⊢ (𝐽 = 𝐼 → ⦋𝐽 / 𝑎⦌𝐴 = ⦋𝐼 / 𝑎⦌𝐴)
46	45	eqcoms 2212	. . . . . 6 ⊢ (𝐼 = 𝐽 → ⦋𝐽 / 𝑎⦌𝐴 = ⦋𝐼 / 𝑎⦌𝐴)
47		eqeq12 2222	. . . . . . 7 ⊢ ((𝑋 = ⦋𝐽 / 𝑎⦌𝐴 ∧ 𝑌 = ⦋𝐼 / 𝑎⦌𝐴) → (𝑋 = 𝑌 ↔ ⦋𝐽 / 𝑎⦌𝐴 = ⦋𝐼 / 𝑎⦌𝐴))
48	47	ancoms 268	. . . . . 6 ⊢ ((𝑌 = ⦋𝐼 / 𝑎⦌𝐴 ∧ 𝑋 = ⦋𝐽 / 𝑎⦌𝐴) → (𝑋 = 𝑌 ↔ ⦋𝐽 / 𝑎⦌𝐴 = ⦋𝐼 / 𝑎⦌𝐴))
49	46, 48	syl5ibrcom 157	. . . . 5 ⊢ (𝐼 = 𝐽 → ((𝑌 = ⦋𝐼 / 𝑎⦌𝐴 ∧ 𝑋 = ⦋𝐽 / 𝑎⦌𝐴) → 𝑋 = 𝑌))
50	49	expd 258	. . . 4 ⊢ (𝐼 = 𝐽 → (𝑌 = ⦋𝐼 / 𝑎⦌𝐴 → (𝑋 = ⦋𝐽 / 𝑎⦌𝐴 → 𝑋 = 𝑌)))
51	50	adantl 277	. . 3 ⊢ ((𝜑 ∧ 𝐼 = 𝐽) → (𝑌 = ⦋𝐼 / 𝑎⦌𝐴 → (𝑋 = ⦋𝐽 / 𝑎⦌𝐴 → 𝑋 = 𝑌)))
52	44, 51	sylbird 170	. 2 ⊢ ((𝜑 ∧ 𝐼 = 𝐽) → (𝐼 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) → (𝑋 = ⦋𝐽 / 𝑎⦌𝐴 → 𝑋 = 𝑌)))
53	6, 27, 52	mp2d 47	1 ⊢ ((𝜑 ∧ 𝐼 = 𝐽) → 𝑋 = 𝑌)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1375   ∈ wcel 2180  ∃!wreu 2490  ⦋csb 3104  ℩crio 5926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-un 3181  df-in 3183  df-ss 3190  df-sn 3652  df-pr 3653  df-uni 3868  df-iota 5254  df-riota 5927

This theorem is referenced by:  uspgredg2v  15984