MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reusv2lem5 Structured version   Visualization version   GIF version

Theorem reusv2lem5 5002
Description: Lemma for reusv2 5003. (Contributed by NM, 4-Jan-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Assertion
Ref Expression
reusv2lem5 ((∀𝑦𝐵 𝐶𝐴𝐵 ≠ ∅) → (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶
Allowed substitution hint:   𝐶(𝑦)

Proof of Theorem reusv2lem5
StepHypRef Expression
1 tru 1635 . . . . . . . . 9
2 biimt 349 . . . . . . . . 9 ((𝐶𝐴 ∧ ⊤) → (𝑥 = 𝐶 ↔ ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶)))
31, 2mpan2 671 . . . . . . . 8 (𝐶𝐴 → (𝑥 = 𝐶 ↔ ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶)))
4 ibar 518 . . . . . . . 8 (𝐶𝐴 → (𝑥 = 𝐶 ↔ (𝐶𝐴𝑥 = 𝐶)))
53, 4bitr3d 270 . . . . . . 7 (𝐶𝐴 → (((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝐶𝐴𝑥 = 𝐶)))
6 eleq1 2838 . . . . . . . 8 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
76pm5.32ri 565 . . . . . . 7 ((𝑥𝐴𝑥 = 𝐶) ↔ (𝐶𝐴𝑥 = 𝐶))
85, 7syl6bbr 278 . . . . . 6 (𝐶𝐴 → (((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝑥𝐴𝑥 = 𝐶)))
98ralimi 3101 . . . . 5 (∀𝑦𝐵 𝐶𝐴 → ∀𝑦𝐵 (((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝑥𝐴𝑥 = 𝐶)))
10 ralbi 3216 . . . . 5 (∀𝑦𝐵 (((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝑥𝐴𝑥 = 𝐶)) → (∀𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ ∀𝑦𝐵 (𝑥𝐴𝑥 = 𝐶)))
119, 10syl 17 . . . 4 (∀𝑦𝐵 𝐶𝐴 → (∀𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ ∀𝑦𝐵 (𝑥𝐴𝑥 = 𝐶)))
1211eubidv 2638 . . 3 (∀𝑦𝐵 𝐶𝐴 → (∃!𝑥𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ ∃!𝑥𝑦𝐵 (𝑥𝐴𝑥 = 𝐶)))
13 r19.28zv 4207 . . . 4 (𝐵 ≠ ∅ → (∀𝑦𝐵 (𝑥𝐴𝑥 = 𝐶) ↔ (𝑥𝐴 ∧ ∀𝑦𝐵 𝑥 = 𝐶)))
1413eubidv 2638 . . 3 (𝐵 ≠ ∅ → (∃!𝑥𝑦𝐵 (𝑥𝐴𝑥 = 𝐶) ↔ ∃!𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝑥 = 𝐶)))
1512, 14sylan9bb 499 . 2 ((∀𝑦𝐵 𝐶𝐴𝐵 ≠ ∅) → (∃!𝑥𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ ∃!𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝑥 = 𝐶)))
161biantrur 520 . . . . 5 (𝑥 = 𝐶 ↔ (⊤ ∧ 𝑥 = 𝐶))
1716rexbii 3189 . . . 4 (∃𝑦𝐵 𝑥 = 𝐶 ↔ ∃𝑦𝐵 (⊤ ∧ 𝑥 = 𝐶))
1817reubii 3277 . . 3 (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥𝐴𝑦𝐵 (⊤ ∧ 𝑥 = 𝐶))
19 reusv2lem4 5001 . . 3 (∃!𝑥𝐴𝑦𝐵 (⊤ ∧ 𝑥 = 𝐶) ↔ ∃!𝑥𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶))
2018, 19bitri 264 . 2 (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶))
21 df-reu 3068 . 2 (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝑥 = 𝐶))
2215, 20, 213bitr4g 303 1 ((∀𝑦𝐵 𝐶𝐴𝐵 ≠ ∅) → (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wtru 1632  wcel 2145  ∃!weu 2618  wne 2943  wral 3061  wrex 3062  ∃!wreu 3063  c0 4063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923  ax-pow 4974
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-nul 4064
This theorem is referenced by:  reusv2  5003
  Copyright terms: Public domain W3C validator