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 31729
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 31727 . . . 4 𝑔𝐶 Fun 𝑔
5 eqid 2778 . . . . . . . 8 (dom 𝑔 ∩ dom ) = (dom 𝑔 ∩ dom )
61, 2, 3, 5bnj1311 31691 . . . . . . 7 ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
763expia 1111 . . . . . 6 ((𝑅 FrSe 𝐴𝑔𝐶) → (𝐶 → (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom ))))
87ralrimiv 3147 . . . . 5 ((𝑅 FrSe 𝐴𝑔𝐶) → ∀𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
98ralrimiva 3148 . . . 4 (𝑅 FrSe 𝐴 → ∀𝑔𝐶𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
10 biid 253 . . . . 5 (∀𝑔𝐶 Fun 𝑔 ↔ ∀𝑔𝐶 Fun 𝑔)
11 biid 253 . . . . 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 31501 . . . 4 ((∀𝑔𝐶 Fun 𝑔 ∧ ∀𝑔𝐶𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom ))) → Fun 𝐶)
134, 9, 12sylancr 581 . . 3 (𝑅 FrSe 𝐴 → Fun 𝐶)
14 bnj60.4 . . . 4 𝐹 = 𝐶
1514funeqi 6156 . . 3 (Fun 𝐹 ↔ Fun 𝐶)
1613, 15sylibr 226 . 2 (𝑅 FrSe 𝐴 → Fun 𝐹)
171, 2, 3, 14bnj1498 31728 . 2 (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
1816, 17bnj1422 31507 1 (𝑅 FrSe 𝐴𝐹 Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  {cab 2763  wral 3090  wrex 3091  cin 3791  wss 3792  cop 4404   cuni 4671  dom cdm 5355  cres 5357  Fun wfun 6129   Fn wfn 6130  cfv 6135   predc-bnj14 31356   FrSe w-bnj15 31360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-reg 8786  ax-inf2 8835
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-om 7344  df-1o 7843  df-bnj17 31355  df-bnj14 31357  df-bnj13 31359  df-bnj15 31361  df-bnj18 31363  df-bnj19 31365
This theorem is referenced by:  bnj1501  31734  bnj1523  31738
  Copyright terms: Public domain W3C validator