MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  riotaeqimp Structured version   Visualization version   GIF version

Theorem riotaeqimp 6854
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 = (𝑎𝑉 𝑋 = 𝐴)
riotaeqimp.j 𝐽 = (𝑎𝑉 𝑌 = 𝐴)
riotaeqimp.x (𝜑 → ∃!𝑎𝑉 𝑋 = 𝐴)
riotaeqimp.y (𝜑 → ∃!𝑎𝑉 𝑌 = 𝐴)
Assertion
Ref Expression
riotaeqimp ((𝜑𝐼 = 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
StepHypRef Expression
1 riotaeqimp.j . . . . . . 7 𝐽 = (𝑎𝑉 𝑌 = 𝐴)
21eqcomi 2815 . . . . . 6 (𝑎𝑉 𝑌 = 𝐴) = 𝐽
32eqeq2i 2818 . . . . 5 (𝐼 = (𝑎𝑉 𝑌 = 𝐴) ↔ 𝐼 = 𝐽)
43a1i 11 . . . 4 (𝜑 → (𝐼 = (𝑎𝑉 𝑌 = 𝐴) ↔ 𝐼 = 𝐽))
54bicomd 214 . . 3 (𝜑 → (𝐼 = 𝐽𝐼 = (𝑎𝑉 𝑌 = 𝐴)))
65biimpa 464 . 2 ((𝜑𝐼 = 𝐽) → 𝐼 = (𝑎𝑉 𝑌 = 𝐴))
7 riotaeqimp.i . . . . 5 𝐼 = (𝑎𝑉 𝑋 = 𝐴)
87eqeq1i 2811 . . . 4 (𝐼 = 𝐽 ↔ (𝑎𝑉 𝑋 = 𝐴) = 𝐽)
9 riotaeqimp.y . . . . . . 7 (𝜑 → ∃!𝑎𝑉 𝑌 = 𝐴)
10 riotacl 6845 . . . . . . 7 (∃!𝑎𝑉 𝑌 = 𝐴 → (𝑎𝑉 𝑌 = 𝐴) ∈ 𝑉)
119, 10syl 17 . . . . . 6 (𝜑 → (𝑎𝑉 𝑌 = 𝐴) ∈ 𝑉)
121, 11syl5eqel 2889 . . . . 5 (𝜑𝐽𝑉)
13 riotaeqimp.x . . . . 5 (𝜑 → ∃!𝑎𝑉 𝑋 = 𝐴)
14 nfv 2005 . . . . . . 7 𝑎 𝐽𝑉
15 nfcvd 2949 . . . . . . 7 (𝐽𝑉𝑎𝐽)
16 nfcvd 2949 . . . . . . . 8 (𝐽𝑉𝑎𝑋)
1715nfcsb1d 3742 . . . . . . . 8 (𝐽𝑉𝑎𝐽 / 𝑎𝐴)
1816, 17nfeqd 2956 . . . . . . 7 (𝐽𝑉 → Ⅎ𝑎 𝑋 = 𝐽 / 𝑎𝐴)
19 id 22 . . . . . . 7 (𝐽𝑉𝐽𝑉)
20 csbeq1a 3737 . . . . . . . . 9 (𝑎 = 𝐽𝐴 = 𝐽 / 𝑎𝐴)
2120eqeq2d 2816 . . . . . . . 8 (𝑎 = 𝐽 → (𝑋 = 𝐴𝑋 = 𝐽 / 𝑎𝐴))
2221adantl 469 . . . . . . 7 ((𝐽𝑉𝑎 = 𝐽) → (𝑋 = 𝐴𝑋 = 𝐽 / 𝑎𝐴))
2314, 15, 18, 19, 22riota2df 6851 . . . . . 6 ((𝐽𝑉 ∧ ∃!𝑎𝑉 𝑋 = 𝐴) → (𝑋 = 𝐽 / 𝑎𝐴 ↔ (𝑎𝑉 𝑋 = 𝐴) = 𝐽))
2423bicomd 214 . . . . 5 ((𝐽𝑉 ∧ ∃!𝑎𝑉 𝑋 = 𝐴) → ((𝑎𝑉 𝑋 = 𝐴) = 𝐽𝑋 = 𝐽 / 𝑎𝐴))
2512, 13, 24syl2anc 575 . . . 4 (𝜑 → ((𝑎𝑉 𝑋 = 𝐴) = 𝐽𝑋 = 𝐽 / 𝑎𝐴))
268, 25syl5bb 274 . . 3 (𝜑 → (𝐼 = 𝐽𝑋 = 𝐽 / 𝑎𝐴))
2726biimpa 464 . 2 ((𝜑𝐼 = 𝐽) → 𝑋 = 𝐽 / 𝑎𝐴)
28 riotacl 6845 . . . . . . . 8 (∃!𝑎𝑉 𝑋 = 𝐴 → (𝑎𝑉 𝑋 = 𝐴) ∈ 𝑉)
2913, 28syl 17 . . . . . . 7 (𝜑 → (𝑎𝑉 𝑋 = 𝐴) ∈ 𝑉)
307, 29syl5eqel 2889 . . . . . 6 (𝜑𝐼𝑉)
31 nfv 2005 . . . . . . 7 𝑎 𝐼𝑉
32 nfcvd 2949 . . . . . . 7 (𝐼𝑉𝑎𝐼)
33 nfcvd 2949 . . . . . . . 8 (𝐼𝑉𝑎𝑌)
3432nfcsb1d 3742 . . . . . . . 8 (𝐼𝑉𝑎𝐼 / 𝑎𝐴)
3533, 34nfeqd 2956 . . . . . . 7 (𝐼𝑉 → Ⅎ𝑎 𝑌 = 𝐼 / 𝑎𝐴)
36 id 22 . . . . . . 7 (𝐼𝑉𝐼𝑉)
37 csbeq1a 3737 . . . . . . . . 9 (𝑎 = 𝐼𝐴 = 𝐼 / 𝑎𝐴)
3837eqeq2d 2816 . . . . . . . 8 (𝑎 = 𝐼 → (𝑌 = 𝐴𝑌 = 𝐼 / 𝑎𝐴))
3938adantl 469 . . . . . . 7 ((𝐼𝑉𝑎 = 𝐼) → (𝑌 = 𝐴𝑌 = 𝐼 / 𝑎𝐴))
4031, 32, 35, 36, 39riota2df 6851 . . . . . 6 ((𝐼𝑉 ∧ ∃!𝑎𝑉 𝑌 = 𝐴) → (𝑌 = 𝐼 / 𝑎𝐴 ↔ (𝑎𝑉 𝑌 = 𝐴) = 𝐼))
4130, 9, 40syl2anc 575 . . . . 5 (𝜑 → (𝑌 = 𝐼 / 𝑎𝐴 ↔ (𝑎𝑉 𝑌 = 𝐴) = 𝐼))
42 eqcom 2813 . . . . 5 ((𝑎𝑉 𝑌 = 𝐴) = 𝐼𝐼 = (𝑎𝑉 𝑌 = 𝐴))
4341, 42syl6bb 278 . . . 4 (𝜑 → (𝑌 = 𝐼 / 𝑎𝐴𝐼 = (𝑎𝑉 𝑌 = 𝐴)))
4443adantr 468 . . 3 ((𝜑𝐼 = 𝐽) → (𝑌 = 𝐼 / 𝑎𝐴𝐼 = (𝑎𝑉 𝑌 = 𝐴)))
45 csbeq1 3731 . . . . . . 7 (𝐽 = 𝐼𝐽 / 𝑎𝐴 = 𝐼 / 𝑎𝐴)
4645eqcoms 2814 . . . . . 6 (𝐼 = 𝐽𝐽 / 𝑎𝐴 = 𝐼 / 𝑎𝐴)
47 eqeq12 2819 . . . . . . 7 ((𝑋 = 𝐽 / 𝑎𝐴𝑌 = 𝐼 / 𝑎𝐴) → (𝑋 = 𝑌𝐽 / 𝑎𝐴 = 𝐼 / 𝑎𝐴))
4847ancoms 448 . . . . . 6 ((𝑌 = 𝐼 / 𝑎𝐴𝑋 = 𝐽 / 𝑎𝐴) → (𝑋 = 𝑌𝐽 / 𝑎𝐴 = 𝐼 / 𝑎𝐴))
4946, 48syl5ibrcom 238 . . . . 5 (𝐼 = 𝐽 → ((𝑌 = 𝐼 / 𝑎𝐴𝑋 = 𝐽 / 𝑎𝐴) → 𝑋 = 𝑌))
5049expd 402 . . . 4 (𝐼 = 𝐽 → (𝑌 = 𝐼 / 𝑎𝐴 → (𝑋 = 𝐽 / 𝑎𝐴𝑋 = 𝑌)))
5150adantl 469 . . 3 ((𝜑𝐼 = 𝐽) → (𝑌 = 𝐼 / 𝑎𝐴 → (𝑋 = 𝐽 / 𝑎𝐴𝑋 = 𝑌)))
5244, 51sylbird 251 . 2 ((𝜑𝐼 = 𝐽) → (𝐼 = (𝑎𝑉 𝑌 = 𝐴) → (𝑋 = 𝐽 / 𝑎𝐴𝑋 = 𝑌)))
536, 27, 52mp2d 49 1 ((𝜑𝐼 = 𝐽) → 𝑋 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2156  ∃!wreu 3098  csb 3728  crio 6830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-un 3774  df-in 3776  df-ss 3783  df-sn 4371  df-pr 4373  df-uni 4631  df-iota 6060  df-riota 6831
This theorem is referenced by:  uspgredg2v  26330
  Copyright terms: Public domain W3C validator