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Theorem wfrresex 8156
Description: Show without using the axiom of replacement that the restriction of the well-ordered recursion generator to a predecessor class is a set. (Contributed by Scott Fenton, 18-Nov-2024.)
Hypothesis
Ref Expression
wfrfun.1 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
wfrresex (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V)

Proof of Theorem wfrresex
StepHypRef Expression
1 wefr 5580 . . . 4 (𝑅 We 𝐴𝑅 Fr 𝐴)
21adantr 481 . . 3 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 Fr 𝐴)
3 weso 5581 . . . . 5 (𝑅 We 𝐴𝑅 Or 𝐴)
4 sopo 5523 . . . . 5 (𝑅 Or 𝐴𝑅 Po 𝐴)
53, 4syl 17 . . . 4 (𝑅 We 𝐴𝑅 Po 𝐴)
65adantr 481 . . 3 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 Po 𝐴)
7 simpr 485 . . 3 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 Se 𝐴)
82, 6, 73jca 1127 . 2 ((𝑅 We 𝐴𝑅 Se 𝐴) → (𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴))
9 wfrfun.1 . . . 4 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
10 df-wrecs 8120 . . . 4 wrecs(𝑅, 𝐴, 𝐺) = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd ))
119, 10eqtri 2768 . . 3 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd ))
1211fprresex 8118 . 2 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V)
138, 12sylan 580 1 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1542  wcel 2110  Vcvv 3431   Po wpo 5502   Or wor 5503   Fr wfr 5542   Se wse 5543   We wwe 5544  dom cdm 5590  cres 5592  ccom 5594  Predcpred 6200  2nd c2nd 7824  frecscfrecs 8088  wrecscwrecs 8119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-po 5504  df-so 5505  df-fr 5545  df-se 5546  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-iota 6390  df-fun 6434  df-fn 6435  df-fv 6440  df-ov 7275  df-frecs 8089  df-wrecs 8120
This theorem is referenced by:  wfr2a  8157
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