MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  riota2df Structured version   Visualization version   GIF version

Theorem riota2df 6773
Description: A deduction version of riota2f 6774. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
riota2df.1 𝑥𝜑
riota2df.2 (𝜑𝑥𝐵)
riota2df.3 (𝜑 → Ⅎ𝑥𝜒)
riota2df.4 (𝜑𝐵𝐴)
riota2df.5 ((𝜑𝑥 = 𝐵) → (𝜓𝜒))
Assertion
Ref Expression
riota2df ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝜒 ↔ (𝑥𝐴 𝜓) = 𝐵))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑥)   𝐵(𝑥)

Proof of Theorem riota2df
StepHypRef Expression
1 riota2df.4 . . . 4 (𝜑𝐵𝐴)
21adantr 466 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → 𝐵𝐴)
3 simpr 471 . . . 4 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → ∃!𝑥𝐴 𝜓)
4 df-reu 3067 . . . 4 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥(𝑥𝐴𝜓))
53, 4sylib 208 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → ∃!𝑥(𝑥𝐴𝜓))
6 simpr 471 . . . . . 6 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
72adantr 466 . . . . . 6 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝐵𝐴)
86, 7eqeltrd 2849 . . . . 5 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝑥𝐴)
98biantrurd 516 . . . 4 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → (𝜓 ↔ (𝑥𝐴𝜓)))
10 riota2df.5 . . . . 5 ((𝜑𝑥 = 𝐵) → (𝜓𝜒))
1110adantlr 686 . . . 4 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → (𝜓𝜒))
129, 11bitr3d 270 . . 3 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → ((𝑥𝐴𝜓) ↔ 𝜒))
13 riota2df.1 . . . 4 𝑥𝜑
14 nfreu1 3257 . . . 4 𝑥∃!𝑥𝐴 𝜓
1513, 14nfan 1979 . . 3 𝑥(𝜑 ∧ ∃!𝑥𝐴 𝜓)
16 riota2df.3 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
1716adantr 466 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → Ⅎ𝑥𝜒)
18 riota2df.2 . . . 4 (𝜑𝑥𝐵)
1918adantr 466 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → 𝑥𝐵)
202, 5, 12, 15, 17, 19iota2df 6018 . 2 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝜒 ↔ (℩𝑥(𝑥𝐴𝜓)) = 𝐵))
21 df-riota 6753 . . 3 (𝑥𝐴 𝜓) = (℩𝑥(𝑥𝐴𝜓))
2221eqeq1i 2775 . 2 ((𝑥𝐴 𝜓) = 𝐵 ↔ (℩𝑥(𝑥𝐴𝜓)) = 𝐵)
2320, 22syl6bbr 278 1 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝜒 ↔ (𝑥𝐴 𝜓) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wnf 1855  wcel 2144  ∃!weu 2617  wnfc 2899  ∃!wreu 3062  cio 5992  crio 6752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-reu 3067  df-v 3351  df-sbc 3586  df-un 3726  df-sn 4315  df-pr 4317  df-uni 4573  df-iota 5994  df-riota 6753
This theorem is referenced by:  riota2f  6774  riotaeqimp  6776  riota5f  6778  mapdheq  37531  hdmap1eq  37604  hdmapval2lem  37634
  Copyright terms: Public domain W3C validator