Users' Mathboxes Mathbox for Emmett Weisz < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  setrec2lem1 Structured version   Visualization version   GIF version

Theorem setrec2lem1 47838
Description: Lemma for setrec2 47840. The functional part of 𝐹 has the same values as 𝐹. (Contributed by Emmett Weisz, 4-Mar-2021.) (New usage is discouraged.)
Assertion
Ref Expression
setrec2lem1 ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹𝑎)
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝑎,𝑦
Allowed substitution hint:   𝐹(𝑎)

Proof of Theorem setrec2lem1
StepHypRef Expression
1 fvres 6910 . 2 (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹𝑎))
2 nfvres 6932 . . 3 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = ∅)
3 vex 3477 . . . . 5 𝑎 ∈ V
4 breq1 5151 . . . . . 6 (𝑥 = 𝑎 → (𝑥𝐹𝑦𝑎𝐹𝑦))
54eubidv 2579 . . . . 5 (𝑥 = 𝑎 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝑎𝐹𝑦))
63, 5elab 3668 . . . 4 (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ∃!𝑦 𝑎𝐹𝑦)
7 tz6.12-2 6879 . . . 4 (¬ ∃!𝑦 𝑎𝐹𝑦 → (𝐹𝑎) = ∅)
86, 7sylnbi 330 . . 3 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → (𝐹𝑎) = ∅)
92, 8eqtr4d 2774 . 2 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹𝑎))
101, 9pm2.61i 182 1 ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹𝑎)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2105  ∃!weu 2561  {cab 2708  c0 4322   class class class wbr 5148  cres 5678  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  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-opab 5211  df-xp 5682  df-dm 5686  df-res 5688  df-iota 6495  df-fv 6551
This theorem is referenced by:  setrec2  47840
  Copyright terms: Public domain W3C validator