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Theorem reusv2lem5 5071
Description: Lemma for reusv2 5072. (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 1642 . . . . . . . . 9
2 biimt 351 . . . . . . . . 9 ((𝐶𝐴 ∧ ⊤) → (𝑥 = 𝐶 ↔ ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶)))
31, 2mpan2 674 . . . . . . . 8 (𝐶𝐴 → (𝑥 = 𝐶 ↔ ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶)))
4 ibar 520 . . . . . . . 8 (𝐶𝐴 → (𝑥 = 𝐶 ↔ (𝐶𝐴𝑥 = 𝐶)))
53, 4bitr3d 272 . . . . . . 7 (𝐶𝐴 → (((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝐶𝐴𝑥 = 𝐶)))
6 eleq1 2873 . . . . . . . 8 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
76pm5.32ri 567 . . . . . . 7 ((𝑥𝐴𝑥 = 𝐶) ↔ (𝐶𝐴𝑥 = 𝐶))
85, 7syl6bbr 280 . . . . . 6 (𝐶𝐴 → (((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝑥𝐴𝑥 = 𝐶)))
98ralimi 3140 . . . . 5 (∀𝑦𝐵 𝐶𝐴 → ∀𝑦𝐵 (((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝑥𝐴𝑥 = 𝐶)))
10 ralbi 3256 . . . . 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 2636 . . 3 (∀𝑦𝐵 𝐶𝐴 → (∃!𝑥𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ ∃!𝑥𝑦𝐵 (𝑥𝐴𝑥 = 𝐶)))
13 r19.28zv 4261 . . . 4 (𝐵 ≠ ∅ → (∀𝑦𝐵 (𝑥𝐴𝑥 = 𝐶) ↔ (𝑥𝐴 ∧ ∀𝑦𝐵 𝑥 = 𝐶)))
1413eubidv 2636 . . 3 (𝐵 ≠ ∅ → (∃!𝑥𝑦𝐵 (𝑥𝐴𝑥 = 𝐶) ↔ ∃!𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝑥 = 𝐶)))
1512, 14sylan9bb 501 . 2 ((∀𝑦𝐵 𝐶𝐴𝐵 ≠ ∅) → (∃!𝑥𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ ∃!𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝑥 = 𝐶)))
161biantrur 522 . . . . 5 (𝑥 = 𝐶 ↔ (⊤ ∧ 𝑥 = 𝐶))
1716rexbii 3229 . . . 4 (∃𝑦𝐵 𝑥 = 𝐶 ↔ ∃𝑦𝐵 (⊤ ∧ 𝑥 = 𝐶))
1817reubii 3317 . . 3 (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥𝐴𝑦𝐵 (⊤ ∧ 𝑥 = 𝐶))
19 reusv2lem4 5070 . . 3 (∃!𝑥𝐴𝑦𝐵 (⊤ ∧ 𝑥 = 𝐶) ↔ ∃!𝑥𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶))
2018, 19bitri 266 . 2 (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶))
21 df-reu 3103 . 2 (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝑥 = 𝐶))
2215, 20, 213bitr4g 305 1 ((∀𝑦𝐵 𝐶𝐴𝐵 ≠ ∅) → (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wtru 1638  wcel 2156  ∃!weu 2630  wne 2978  wral 3096  wrex 3097  ∃!wreu 3098  c0 4116
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-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-nul 4983  ax-pow 5035
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-nul 4117
This theorem is referenced by:  reusv2  5072
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