Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frrlem5d Structured version   Visualization version   GIF version

Theorem frrlem5d 31771
Description: Lemma for founded recursion. The domain of the union of a subset of 𝐵 is a subset of 𝐴. (Contributed by Paul Chapman, 29-Apr-2012.)
Hypotheses
Ref Expression
frrlem5.1 𝑅 Fr 𝐴
frrlem5.2 𝑅 Se 𝐴
frrlem5.3 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}
Assertion
Ref Expression
frrlem5d (𝐶𝐵 → dom 𝐶𝐴)
Distinct variable groups:   𝐴,𝑓,𝑥,𝑦   𝑓,𝐺,𝑥,𝑦   𝑅,𝑓,𝑥,𝑦   𝑥,𝐵
Allowed substitution hints:   𝐵(𝑦,𝑓)   𝐶(𝑥,𝑦,𝑓)

Proof of Theorem frrlem5d
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 dmuni 5332 . 2 dom 𝐶 = 𝑔𝐶 dom 𝑔
2 ssel 3595 . . . . 5 (𝐶𝐵 → (𝑔𝐶𝑔𝐵))
3 frrlem5.3 . . . . . 6 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}
43frrlem3 31766 . . . . 5 (𝑔𝐵 → dom 𝑔𝐴)
52, 4syl6 35 . . . 4 (𝐶𝐵 → (𝑔𝐶 → dom 𝑔𝐴))
65ralrimiv 2964 . . 3 (𝐶𝐵 → ∀𝑔𝐶 dom 𝑔𝐴)
7 iunss 4559 . . 3 ( 𝑔𝐶 dom 𝑔𝐴 ↔ ∀𝑔𝐶 dom 𝑔𝐴)
86, 7sylibr 224 . 2 (𝐶𝐵 𝑔𝐶 dom 𝑔𝐴)
91, 8syl5eqss 3647 1 (𝐶𝐵 → dom 𝐶𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1037   = wceq 1482  wex 1703  wcel 1989  {cab 2607  wral 2911  wss 3572   cuni 4434   ciun 4518   Fr wfr 5068   Se wse 5069  dom cdm 5112  cres 5114  Predcpred 5677   Fn wfn 5881  cfv 5886  (class class class)co 6647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-iota 5849  df-fun 5888  df-fn 5889  df-fv 5894  df-ov 6650
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator