Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj60 Structured version   Visualization version   GIF version

Theorem bnj60 31115
Description: Well-founded recursion, part 1 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj60.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj60.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj60.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj60.4 𝐹 = 𝐶
Assertion
Ref Expression
bnj60 (𝑅 FrSe 𝐴𝐹 Fn 𝐴)
Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝐵,𝑓   𝐺,𝑑,𝑓,𝑥   𝑅,𝑑,𝑓,𝑥
Allowed substitution hints:   𝐵(𝑥,𝑑)   𝐶(𝑥,𝑓,𝑑)   𝐹(𝑥,𝑓,𝑑)   𝑌(𝑥,𝑓,𝑑)

Proof of Theorem bnj60
Dummy variables 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj60.1 . . . . 5 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
2 bnj60.2 . . . . 5 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
3 bnj60.3 . . . . 5 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
41, 2, 3bnj1497 31113 . . . 4 𝑔𝐶 Fun 𝑔
5 eqid 2621 . . . . . . . 8 (dom 𝑔 ∩ dom ) = (dom 𝑔 ∩ dom )
61, 2, 3, 5bnj1311 31077 . . . . . . 7 ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
763expia 1266 . . . . . 6 ((𝑅 FrSe 𝐴𝑔𝐶) → (𝐶 → (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom ))))
87ralrimiv 2964 . . . . 5 ((𝑅 FrSe 𝐴𝑔𝐶) → ∀𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
98ralrimiva 2965 . . . 4 (𝑅 FrSe 𝐴 → ∀𝑔𝐶𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
10 biid 251 . . . . 5 (∀𝑔𝐶 Fun 𝑔 ↔ ∀𝑔𝐶 Fun 𝑔)
11 biid 251 . . . . 5 ((∀𝑔𝐶 Fun 𝑔 ∧ ∀𝑔𝐶𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom ))) ↔ (∀𝑔𝐶 Fun 𝑔 ∧ ∀𝑔𝐶𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom ))))
1210, 5, 11bnj1383 30887 . . . 4 ((∀𝑔𝐶 Fun 𝑔 ∧ ∀𝑔𝐶𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom ))) → Fun 𝐶)
134, 9, 12sylancr 695 . . 3 (𝑅 FrSe 𝐴 → Fun 𝐶)
14 bnj60.4 . . . 4 𝐹 = 𝐶
1514funeqi 5907 . . 3 (Fun 𝐹 ↔ Fun 𝐶)
1613, 15sylibr 224 . 2 (𝑅 FrSe 𝐴 → Fun 𝐹)
171, 2, 3, 14bnj1498 31114 . 2 (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
1816, 17bnj1422 30893 1 (𝑅 FrSe 𝐴𝐹 Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  {cab 2607  wral 2911  wrex 2912  cin 3571  wss 3572  cop 4181   cuni 4434  dom cdm 5112  cres 5114  Fun wfun 5880   Fn wfn 5881  cfv 5886   predc-bnj14 30739   FrSe w-bnj15 30743
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-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-reg 8494  ax-inf2 8535
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-fal 1488  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  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-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-om 7063  df-1o 7557  df-bnj17 30738  df-bnj14 30740  df-bnj13 30742  df-bnj15 30744  df-bnj18 30746  df-bnj19 30748
This theorem is referenced by:  bnj1501  31120  bnj1523  31124
  Copyright terms: Public domain W3C validator