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Mirrors > Home > MPE Home > Th. List > wfrresex | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
wfrfun.1 | ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) |
Ref | Expression |
---|---|
wfrresex | ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wefr 5580 | . . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴) | |
2 | 1 | adantr 481 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Fr 𝐴) |
3 | weso 5581 | . . . . 5 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴) | |
4 | sopo 5523 | . . . . 5 ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Po 𝐴) |
6 | 5 | adantr 481 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Po 𝐴) |
7 | simpr 485 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Se 𝐴) | |
8 | 2, 6, 7 | 3jca 1127 | . 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → (𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 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 )) | |
11 | 9, 10 | eqtri 2768 | . . 3 ⊢ 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd )) |
12 | 11 | fprresex 8118 | . 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V) |
13 | 8, 12 | sylan 580 | 1 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 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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