Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  reusv1 Structured version   Visualization version   GIF version

Theorem reusv1 5145
 Description: Two ways to express single-valuedness of a class expression 𝐶(𝑦). (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) (Proof shortened by JJ, 7-Aug-2021.)
Assertion
Ref Expression
reusv1 (∃𝑦𝐵 𝜑 → (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑦)   𝐶(𝑦)

Proof of Theorem reusv1
StepHypRef Expression
1 nftru 1767 . . . . 5 𝑥
2 nfra1 3163 . . . . . 6 𝑦𝑦𝐵 (𝜑𝑥 = 𝐶)
32a1i 11 . . . . 5 (⊤ → Ⅎ𝑦𝑦𝐵 (𝜑𝑥 = 𝐶))
41, 3nfmodv 2572 . . . 4 (⊤ → Ⅎ𝑦∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶))
54mptru 1514 . . 3 𝑦∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶)
6 rsp 3149 . . . . . 6 (∀𝑦𝐵 (𝜑𝑥 = 𝐶) → (𝑦𝐵 → (𝜑𝑥 = 𝐶)))
76com3l 89 . . . . 5 (𝑦𝐵 → (𝜑 → (∀𝑦𝐵 (𝜑𝑥 = 𝐶) → 𝑥 = 𝐶)))
87alrimdv 1888 . . . 4 (𝑦𝐵 → (𝜑 → ∀𝑥(∀𝑦𝐵 (𝜑𝑥 = 𝐶) → 𝑥 = 𝐶)))
9 mo2icl 3613 . . . 4 (∀𝑥(∀𝑦𝐵 (𝜑𝑥 = 𝐶) → 𝑥 = 𝐶) → ∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶))
108, 9syl6 35 . . 3 (𝑦𝐵 → (𝜑 → ∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶)))
115, 10rexlimi 3252 . 2 (∃𝑦𝐵 𝜑 → ∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶))
12 mormo 3363 . 2 (∃*𝑥𝑦𝐵 (𝜑𝑥 = 𝐶) → ∃*𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))
13 reu5 3364 . . 3 (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ (∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ∧ ∃*𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
1413rbaib 531 . 2 (∃*𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) → (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
1511, 12, 143syl 18 1 (∃𝑦𝐵 𝜑 → (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1505   = wceq 1507  ⊤wtru 1508  Ⅎwnf 1746   ∈ wcel 2050  ∃*wmo 2545  ∀wral 3082  ∃wrex 3083  ∃!wreu 3084  ∃*wrmo 3085 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-v 3411 This theorem is referenced by:  cdleme25c  36936  cdleme29c  36957  cdlemefrs29cpre1  36979  cdlemk29-3  37492  cdlemkid5  37516  dihlsscpre  37815  mapdh9a  38370  mapdh9aOLDN  38371
 Copyright terms: Public domain W3C validator