Theorem riotaeqimp 7393

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	. . . . 5 ⊢ 𝐽 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴)
2	1	eqcomi 2772	. . . 4 ⊢ (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) = 𝐽
3	2	eqeq2i 2776	. . 3 ⊢ (𝐼 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) ↔ 𝐼 = 𝐽)
4	3	bilanri 511	. 2 ⊢ ((𝜑 ∧ 𝐼 = 𝐽) → 𝐼 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴))
5		riotaeqimp.i	. . . . 5 ⊢ 𝐼 = (℩𝑎 ∈ 𝑉 𝑋 = 𝐴)
6	5	eqeq1i 2768	. . . 4 ⊢ (𝐼 = 𝐽 ↔ (℩𝑎 ∈ 𝑉 𝑋 = 𝐴) = 𝐽)
7		riotaeqimp.y	. . . . . . 7 ⊢ (𝜑 → ∃!𝑎 ∈ 𝑉 𝑌 = 𝐴)
8		riotacl 7384	. . . . . . 7 ⊢ (∃!𝑎 ∈ 𝑉 𝑌 = 𝐴 → (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) ∈ 𝑉)
9	7, 8	syl 18	. . . . . 6 ⊢ (𝜑 → (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) ∈ 𝑉)
10	1, 9	eqeltrid 2867	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ 𝑉)
11		riotaeqimp.x	. . . . 5 ⊢ (𝜑 → ∃!𝑎 ∈ 𝑉 𝑋 = 𝐴)
12		nfv 1944	. . . . . . 7 ⊢ Ⅎ𝑎 𝐽 ∈ 𝑉
13		nfcvd 2926	. . . . . . 7 ⊢ (𝐽 ∈ 𝑉 → Ⅎ𝑎𝐽)
14		nfcvd 2926	. . . . . . . 8 ⊢ (𝐽 ∈ 𝑉 → Ⅎ𝑎𝑋)
15	13	nfcsb1d 3875	. . . . . . . 8 ⊢ (𝐽 ∈ 𝑉 → Ⅎ𝑎⦋𝐽 / 𝑎⦌𝐴)
16	14, 15	nfeqd 2935	. . . . . . 7 ⊢ (𝐽 ∈ 𝑉 → Ⅎ𝑎 𝑋 = ⦋𝐽 / 𝑎⦌𝐴)
17		id 23	. . . . . . 7 ⊢ (𝐽 ∈ 𝑉 → 𝐽 ∈ 𝑉)
18		csbeq1a 3867	. . . . . . . . 9 ⊢ (𝑎 = 𝐽 → 𝐴 = ⦋𝐽 / 𝑎⦌𝐴)
19	18	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑎 = 𝐽 → (𝑋 = 𝐴 ↔ 𝑋 = ⦋𝐽 / 𝑎⦌𝐴))
20	19	adantl 486	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝑎 = 𝐽) → (𝑋 = 𝐴 ↔ 𝑋 = ⦋𝐽 / 𝑎⦌𝐴))
21	12, 13, 16, 17, 20	riota2df 7390	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ ∃!𝑎 ∈ 𝑉 𝑋 = 𝐴) → (𝑋 = ⦋𝐽 / 𝑎⦌𝐴 ↔ (℩𝑎 ∈ 𝑉 𝑋 = 𝐴) = 𝐽))
22	21	bicomd 226	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ ∃!𝑎 ∈ 𝑉 𝑋 = 𝐴) → ((℩𝑎 ∈ 𝑉 𝑋 = 𝐴) = 𝐽 ↔ 𝑋 = ⦋𝐽 / 𝑎⦌𝐴))
23	10, 11, 22	syl2anc 595	. . . 4 ⊢ (𝜑 → ((℩𝑎 ∈ 𝑉 𝑋 = 𝐴) = 𝐽 ↔ 𝑋 = ⦋𝐽 / 𝑎⦌𝐴))
24	6, 23	bitrid 286	. . 3 ⊢ (𝜑 → (𝐼 = 𝐽 ↔ 𝑋 = ⦋𝐽 / 𝑎⦌𝐴))
25	24	biimpa 481	. 2 ⊢ ((𝜑 ∧ 𝐼 = 𝐽) → 𝑋 = ⦋𝐽 / 𝑎⦌𝐴)
26		riotacl 7384	. . . . . . . 8 ⊢ (∃!𝑎 ∈ 𝑉 𝑋 = 𝐴 → (℩𝑎 ∈ 𝑉 𝑋 = 𝐴) ∈ 𝑉)
27	11, 26	syl 18	. . . . . . 7 ⊢ (𝜑 → (℩𝑎 ∈ 𝑉 𝑋 = 𝐴) ∈ 𝑉)
28	5, 27	eqeltrid 2867	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
29		nfv 1944	. . . . . . 7 ⊢ Ⅎ𝑎 𝐼 ∈ 𝑉
30		nfcvd 2926	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → Ⅎ𝑎𝐼)
31		nfcvd 2926	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → Ⅎ𝑎𝑌)
32	30	nfcsb1d 3875	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → Ⅎ𝑎⦋𝐼 / 𝑎⦌𝐴)
33	31, 32	nfeqd 2935	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → Ⅎ𝑎 𝑌 = ⦋𝐼 / 𝑎⦌𝐴)
34		id 23	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ 𝑉)
35		csbeq1a 3867	. . . . . . . . 9 ⊢ (𝑎 = 𝐼 → 𝐴 = ⦋𝐼 / 𝑎⦌𝐴)
36	35	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑎 = 𝐼 → (𝑌 = 𝐴 ↔ 𝑌 = ⦋𝐼 / 𝑎⦌𝐴))
37	36	adantl 486	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑎 = 𝐼) → (𝑌 = 𝐴 ↔ 𝑌 = ⦋𝐼 / 𝑎⦌𝐴))
38	29, 30, 33, 34, 37	riota2df 7390	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ ∃!𝑎 ∈ 𝑉 𝑌 = 𝐴) → (𝑌 = ⦋𝐼 / 𝑎⦌𝐴 ↔ (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) = 𝐼))
39	28, 7, 38	syl2anc 595	. . . . 5 ⊢ (𝜑 → (𝑌 = ⦋𝐼 / 𝑎⦌𝐴 ↔ (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) = 𝐼))
40		eqcom 2770	. . . . 5 ⊢ ((℩𝑎 ∈ 𝑉 𝑌 = 𝐴) = 𝐼 ↔ 𝐼 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴))
41	39, 40	bitrdi 290	. . . 4 ⊢ (𝜑 → (𝑌 = ⦋𝐼 / 𝑎⦌𝐴 ↔ 𝐼 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴)))
42	41	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝐼 = 𝐽) → (𝑌 = ⦋𝐼 / 𝑎⦌𝐴 ↔ 𝐼 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴)))
43		csbeq1 3856	. . . . . . 7 ⊢ (𝐽 = 𝐼 → ⦋𝐽 / 𝑎⦌𝐴 = ⦋𝐼 / 𝑎⦌𝐴)
44	43	eqcoms 2771	. . . . . 6 ⊢ (𝐼 = 𝐽 → ⦋𝐽 / 𝑎⦌𝐴 = ⦋𝐼 / 𝑎⦌𝐴)
45		eqeq12 2780	. . . . . . 7 ⊢ ((𝑋 = ⦋𝐽 / 𝑎⦌𝐴 ∧ 𝑌 = ⦋𝐼 / 𝑎⦌𝐴) → (𝑋 = 𝑌 ↔ ⦋𝐽 / 𝑎⦌𝐴 = ⦋𝐼 / 𝑎⦌𝐴))
46	45	ancoms 463	. . . . . 6 ⊢ ((𝑌 = ⦋𝐼 / 𝑎⦌𝐴 ∧ 𝑋 = ⦋𝐽 / 𝑎⦌𝐴) → (𝑋 = 𝑌 ↔ ⦋𝐽 / 𝑎⦌𝐴 = ⦋𝐼 / 𝑎⦌𝐴))
47	44, 46	syl5ibrcom 250	. . . . 5 ⊢ (𝐼 = 𝐽 → ((𝑌 = ⦋𝐼 / 𝑎⦌𝐴 ∧ 𝑋 = ⦋𝐽 / 𝑎⦌𝐴) → 𝑋 = 𝑌))
48	47	expd 420	. . . 4 ⊢ (𝐼 = 𝐽 → (𝑌 = ⦋𝐼 / 𝑎⦌𝐴 → (𝑋 = ⦋𝐽 / 𝑎⦌𝐴 → 𝑋 = 𝑌)))
49	48	adantl 486	. . 3 ⊢ ((𝜑 ∧ 𝐼 = 𝐽) → (𝑌 = ⦋𝐼 / 𝑎⦌𝐴 → (𝑋 = ⦋𝐽 / 𝑎⦌𝐴 → 𝑋 = 𝑌)))
50	42, 49	sylbird 263	. 2 ⊢ ((𝜑 ∧ 𝐼 = 𝐽) → (𝐼 = (℩𝑎 ∈ 𝑉 𝑌 = 𝐴) → (𝑋 = ⦋𝐽 / 𝑎⦌𝐴 → 𝑋 = 𝑌)))
51	4, 25, 50	mp2d 50	1 ⊢ ((𝜑 ∧ 𝐼 = 𝐽) → 𝑋 = 𝑌)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃!wreu 3367  ⦋csb 3853  ℩crio 7366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-un 3910  df-ss 3922  df-sn 4590  df-pr 4592  df-uni 4873  df-iota 6492  df-riota 7367

This theorem is referenced by:  uspgredg2v  29574