Users' Mathboxes 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 41874
Description: Lemma for fnwe2 41877. 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 3146 . . 3 (𝜑 → ∀𝑥𝐴 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
3 fveq2 6891 . . . . . . 7 (𝑎 = 𝑥 → (𝐹𝑎) = (𝐹𝑥))
43csbeq1d 3897 . . . . . 6 (𝑎 = 𝑥(𝐹𝑎) / 𝑧𝑆 = (𝐹𝑥) / 𝑧𝑆)
5 fvex 6904 . . . . . . 7 (𝐹𝑥) ∈ V
6 fnwe2.su . . . . . . 7 (𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)
75, 6csbie 3929 . . . . . 6 (𝐹𝑥) / 𝑧𝑆 = 𝑈
84, 7eqtrdi 2788 . . . . 5 (𝑎 = 𝑥(𝐹𝑎) / 𝑧𝑆 = 𝑈)
93eqeq2d 2743 . . . . . 6 (𝑎 = 𝑥 → ((𝐹𝑦) = (𝐹𝑎) ↔ (𝐹𝑦) = (𝐹𝑥)))
109rabbidv 3440 . . . . 5 (𝑎 = 𝑥 → {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)} = {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
118, 10weeq12d 41864 . . . 4 (𝑎 = 𝑥 → ((𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)} ↔ 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)}))
1211cbvralvw 3234 . . 3 (∀𝑎𝐴 (𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)} ↔ ∀𝑥𝐴 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
132, 12sylibr 233 . 2 (𝜑 → ∀𝑎𝐴 (𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
1413r19.21bi 3248 1 ((𝜑𝑎𝐴) → (𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 845   = wceq 1541  wcel 2106  wral 3061  {crab 3432  csb 3893   class class class wbr 5148  {copab 5210   We wwe 5630  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-iota 6495  df-fv 6551
This theorem is referenced by:  fnwe2lem2  41875  fnwe2lem3  41876
  Copyright terms: Public domain W3C validator