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Theorem wfrfun 8328
Description: The "function" generated by the well-ordered recursion generator is indeed a function. Avoids the axiom of replacement. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 17-Nov-2024.)
Hypothesis
Ref Expression
wfrfun.1 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
wfrfun ((𝑅 We 𝐴𝑅 Se 𝐴) → Fun 𝐹)

Proof of Theorem wfrfun
StepHypRef Expression
1 wefr 5657 . . 3 (𝑅 We 𝐴𝑅 Fr 𝐴)
21adantr 480 . 2 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 Fr 𝐴)
3 weso 5658 . . . 4 (𝑅 We 𝐴𝑅 Or 𝐴)
4 sopo 5598 . . . 4 (𝑅 Or 𝐴𝑅 Po 𝐴)
53, 4syl 17 . . 3 (𝑅 We 𝐴𝑅 Po 𝐴)
65adantr 480 . 2 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 Po 𝐴)
7 simpr 484 . 2 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 Se 𝐴)
8 wfrfun.1 . . . 4 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
9 df-wrecs 8293 . . . 4 wrecs(𝑅, 𝐴, 𝐺) = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd ))
108, 9eqtri 2752 . . 3 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd ))
1110fprfung 8290 . 2 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → Fun 𝐹)
122, 6, 7, 11syl3anc 1368 1 ((𝑅 We 𝐴𝑅 Se 𝐴) → Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533   Po wpo 5577   Or wor 5578   Fr wfr 5619   Se wse 5620   We wwe 5621  ccom 5671  Fun wfun 6528  2nd c2nd 7968  frecscfrecs 8261  wrecscwrecs 8292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-po 5579  df-so 5580  df-fr 5622  df-se 5623  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-iota 6486  df-fun 6536  df-fn 6537  df-fv 6542  df-ov 7405  df-frecs 8262  df-wrecs 8293
This theorem is referenced by:  bpolylem  15994
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