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Theorem bnj60 30384
 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 30382 . . . 4 𝑔𝐶 Fun 𝑔
5 eqid 2610 . . . . . . . 8 (dom 𝑔 ∩ dom ) = (dom 𝑔 ∩ dom )
61, 2, 3, 5bnj1311 30346 . . . . . . 7 ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
763expia 1259 . . . . . 6 ((𝑅 FrSe 𝐴𝑔𝐶) → (𝐶 → (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom ))))
87ralrimiv 2948 . . . . 5 ((𝑅 FrSe 𝐴𝑔𝐶) → ∀𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
98ralrimiva 2949 . . . 4 (𝑅 FrSe 𝐴 → ∀𝑔𝐶𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom )))
10 biid 250 . . . . 5 (∀𝑔𝐶 Fun 𝑔 ↔ ∀𝑔𝐶 Fun 𝑔)
11 biid 250 . . . . 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 30156 . . . 4 ((∀𝑔𝐶 Fun 𝑔 ∧ ∀𝑔𝐶𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom )) = ( ↾ (dom 𝑔 ∩ dom ))) → Fun 𝐶)
134, 9, 12sylancr 694 . . 3 (𝑅 FrSe 𝐴 → Fun 𝐶)
14 bnj60.4 . . . 4 𝐹 = 𝐶
1514funeqi 5824 . . 3 (Fun 𝐹 ↔ Fun 𝐶)
1613, 15sylibr 223 . 2 (𝑅 FrSe 𝐴 → Fun 𝐹)
171, 2, 3, 14bnj1498 30383 . 2 (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
1816, 17bnj1422 30162 1 (𝑅 FrSe 𝐴𝐹 Fn 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  ⟨cop 4131  ∪ cuni 4372  dom cdm 5038   ↾ cres 5040  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804   predc-bnj14 30007   FrSe w-bnj15 30011 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-bnj17 30006  df-bnj14 30008  df-bnj13 30010  df-bnj15 30012  df-bnj18 30014  df-bnj19 30016 This theorem is referenced by:  bnj1501  30389  bnj1523  30393
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