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 40360
 Description: Lemma for fnwe2 40363. 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 3114 . . 3 (𝜑 → ∀𝑥𝐴 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
3 fveq2 6659 . . . . . . 7 (𝑎 = 𝑥 → (𝐹𝑎) = (𝐹𝑥))
43csbeq1d 3810 . . . . . 6 (𝑎 = 𝑥(𝐹𝑎) / 𝑧𝑆 = (𝐹𝑥) / 𝑧𝑆)
5 fvex 6672 . . . . . . 7 (𝐹𝑥) ∈ V
6 fnwe2.su . . . . . . 7 (𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)
75, 6csbie 3841 . . . . . 6 (𝐹𝑥) / 𝑧𝑆 = 𝑈
84, 7eqtrdi 2810 . . . . 5 (𝑎 = 𝑥(𝐹𝑎) / 𝑧𝑆 = 𝑈)
93eqeq2d 2770 . . . . . 6 (𝑎 = 𝑥 → ((𝐹𝑦) = (𝐹𝑎) ↔ (𝐹𝑦) = (𝐹𝑥)))
109rabbidv 3393 . . . . 5 (𝑎 = 𝑥 → {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)} = {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
118, 10weeq12d 40350 . . . 4 (𝑎 = 𝑥 → ((𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)} ↔ 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)}))
1211cbvralvw 3362 . . 3 (∀𝑎𝐴 (𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)} ↔ ∀𝑥𝐴 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
132, 12sylibr 237 . 2 (𝜑 → ∀𝑎𝐴 (𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
1413r19.21bi 3138 1 ((𝜑𝑎𝐴) → (𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 845   = wceq 1539   ∈ wcel 2112  ∀wral 3071  {crab 3075  ⦋csb 3806   class class class wbr 5033  {copab 5095   We wwe 5483  ‘cfv 6336 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-nul 5177 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-iota 6295  df-fv 6344 This theorem is referenced by:  fnwe2lem2  40361  fnwe2lem3  40362
 Copyright terms: Public domain W3C validator