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Theorem wfr1 8282
Description: The Principle of Well-Ordered Recursion, part 1 of 3. We start with an arbitrary function 𝐺. Then, using a base class 𝐴 and a set-like well-ordering 𝑅 of 𝐴, we define a function 𝐹. This function is said to be defined by "well-ordered recursion". The purpose of these three theorems is to demonstrate the properties of 𝐹. We begin by showing that 𝐹 is a function over 𝐴. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 18-Nov-2024.)
Hypothesis
Ref Expression
wfr1.1 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
wfr1 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝐹 Fn 𝐴)

Proof of Theorem wfr1
StepHypRef Expression
1 wefr 5624 . . 3 (𝑅 We 𝐴𝑅 Fr 𝐴)
21adantr 482 . 2 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 Fr 𝐴)
3 weso 5625 . . . 4 (𝑅 We 𝐴𝑅 Or 𝐴)
4 sopo 5565 . . . 4 (𝑅 Or 𝐴𝑅 Po 𝐴)
53, 4syl 17 . . 3 (𝑅 We 𝐴𝑅 Po 𝐴)
65adantr 482 . 2 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 Po 𝐴)
7 simpr 486 . 2 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 Se 𝐴)
8 wfr1.1 . . . 4 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
9 df-wrecs 8244 . . . 4 wrecs(𝑅, 𝐴, 𝐺) = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd ))
108, 9eqtri 2761 . . 3 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd ))
1110fpr1 8235 . 2 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → 𝐹 Fn 𝐴)
122, 6, 7, 11syl3anc 1372 1 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝐹 Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542   Po wpo 5544   Or wor 5545   Fr wfr 5586   Se wse 5587   We wwe 5588  ccom 5638   Fn wfn 6492  2nd c2nd 7921  frecscfrecs 8212  wrecscwrecs 8243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-frecs 8213  df-wrecs 8244
This theorem is referenced by:  wfr2  8283  wfr3  8284  wfr3OLD  8285  tfr1ALT  8347  bpolylem  15936
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