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Theorem wfrfun 8353
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 5668 . . 3 (𝑅 We 𝐴𝑅 Fr 𝐴)
21adantr 480 . 2 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 Fr 𝐴)
3 weso 5669 . . . 4 (𝑅 We 𝐴𝑅 Or 𝐴)
4 sopo 5609 . . . 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 8318 . . . 4 wrecs(𝑅, 𝐴, 𝐺) = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd ))
108, 9eqtri 2756 . . 3 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd ))
1110fprfung 8315 . 2 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → Fun 𝐹)
122, 6, 7, 11syl3anc 1369 1 ((𝑅 We 𝐴𝑅 Se 𝐴) → Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534   Po wpo 5588   Or wor 5589   Fr wfr 5630   Se wse 5631   We wwe 5632  ccom 5682  Fun wfun 6542  2nd c2nd 7992  frecscfrecs 8286  wrecscwrecs 8317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-iota 6500  df-fun 6550  df-fn 6551  df-fv 6556  df-ov 7423  df-frecs 8287  df-wrecs 8318
This theorem is referenced by:  bpolylem  16025
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