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

Theorem riota2df 7131
Description: A deduction version of riota2f 7132. (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 481 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → 𝐵𝐴)
3 simpr 485 . . . 4 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → ∃!𝑥𝐴 𝜓)
4 df-reu 3150 . . . 4 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥(𝑥𝐴𝜓))
53, 4sylib 219 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → ∃!𝑥(𝑥𝐴𝜓))
6 simpr 485 . . . . . 6 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
72adantr 481 . . . . . 6 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝐵𝐴)
86, 7eqeltrd 2918 . . . . 5 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝑥𝐴)
98biantrurd 533 . . . 4 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → (𝜓 ↔ (𝑥𝐴𝜓)))
10 riota2df.5 . . . . 5 ((𝜑𝑥 = 𝐵) → (𝜓𝜒))
1110adantlr 711 . . . 4 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → (𝜓𝜒))
129, 11bitr3d 282 . . 3 (((𝜑 ∧ ∃!𝑥𝐴 𝜓) ∧ 𝑥 = 𝐵) → ((𝑥𝐴𝜓) ↔ 𝜒))
13 riota2df.1 . . . 4 𝑥𝜑
14 nfreu1 3376 . . . 4 𝑥∃!𝑥𝐴 𝜓
1513, 14nfan 1893 . . 3 𝑥(𝜑 ∧ ∃!𝑥𝐴 𝜓)
16 riota2df.3 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
1716adantr 481 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → Ⅎ𝑥𝜒)
18 riota2df.2 . . . 4 (𝜑𝑥𝐵)
1918adantr 481 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → 𝑥𝐵)
202, 5, 12, 15, 17, 19iota2df 6341 . 2 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝜒 ↔ (℩𝑥(𝑥𝐴𝜓)) = 𝐵))
21 df-riota 7108 . . 3 (𝑥𝐴 𝜓) = (℩𝑥(𝑥𝐴𝜓))
2221eqeq1i 2831 . 2 ((𝑥𝐴 𝜓) = 𝐵 ↔ (℩𝑥(𝑥𝐴𝜓)) = 𝐵)
2320, 22syl6bbr 290 1 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝜒 ↔ (𝑥𝐴 𝜓) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wnf 1777  wcel 2107  ∃!weu 2651  wnfc 2966  ∃!wreu 3145  cio 6311  crio 7107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-reu 3150  df-v 3502  df-sbc 3777  df-un 3945  df-sn 4565  df-pr 4567  df-uni 4838  df-iota 6313  df-riota 7108
This theorem is referenced by:  riota2f  7132  riotaeqimp  7134  riota5f  7136  mapdheq  38750  hdmap1eq  38823  hdmapval2lem  38853
  Copyright terms: Public domain W3C validator