Mathbox for Stefan O'Rear < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fnwe2lem1 Structured version   Visualization version   GIF version

Theorem fnwe2lem1 36638
 Description: Lemma for fnwe2 36641. Substitution in well-ordering hypothesis. (Contributed by Stefan O'Rear, 19-Jan-2015.)
Hypotheses
Ref Expression
fnwe2.su (𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)
fnwe2.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦))}
fnwe2.s ((𝜑𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
Assertion
Ref Expression
fnwe2lem1 ((𝜑𝑎𝐴) → (𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
Distinct variable groups:   𝑦,𝑈,𝑧,𝑎   𝑥,𝑆,𝑦,𝑎   𝑥,𝑅,𝑦,𝑎   𝜑,𝑥,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧,𝑎   𝑥,𝐹,𝑦,𝑧,𝑎   𝑇,𝑎
Allowed substitution hints:   𝜑(𝑎)   𝑅(𝑧)   𝑆(𝑧)   𝑇(𝑥,𝑦,𝑧)   𝑈(𝑥)

Proof of Theorem fnwe2lem1
StepHypRef Expression
1 fnwe2.s . . . 4 ((𝜑𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
21ralrimiva 2949 . . 3 (𝜑 → ∀𝑥𝐴 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
3 fveq2 6103 . . . . . . 7 (𝑎 = 𝑥 → (𝐹𝑎) = (𝐹𝑥))
43csbeq1d 3506 . . . . . 6 (𝑎 = 𝑥(𝐹𝑎) / 𝑧𝑆 = (𝐹𝑥) / 𝑧𝑆)
5 fvex 6113 . . . . . . 7 (𝐹𝑥) ∈ V
6 fnwe2.su . . . . . . 7 (𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)
75, 6csbie 3525 . . . . . 6 (𝐹𝑥) / 𝑧𝑆 = 𝑈
84, 7syl6eq 2660 . . . . 5 (𝑎 = 𝑥(𝐹𝑎) / 𝑧𝑆 = 𝑈)
93eqeq2d 2620 . . . . . 6 (𝑎 = 𝑥 → ((𝐹𝑦) = (𝐹𝑎) ↔ (𝐹𝑦) = (𝐹𝑥)))
109rabbidv 3164 . . . . 5 (𝑎 = 𝑥 → {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)} = {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
118, 10weeq12d 36628 . . . 4 (𝑎 = 𝑥 → ((𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)} ↔ 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)}))
1211cbvralv 3147 . . 3 (∀𝑎𝐴 (𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)} ↔ ∀𝑥𝐴 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
132, 12sylibr 223 . 2 (𝜑 → ∀𝑎𝐴 (𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
1413r19.21bi 2916 1 ((𝜑𝑎𝐴) → (𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  ⦋csb 3499   class class class wbr 4583  {copab 4642   We wwe 4996  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-iota 5768  df-fv 5812 This theorem is referenced by:  fnwe2lem2  36639  fnwe2lem3  36640
 Copyright terms: Public domain W3C validator