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Mirrors > Home > MPE Home > Th. List > wfrdmcl | Structured version Visualization version GIF version |
Description: The predecessor class of an element of the well-ordered recursion generator's domain is a subset of its domain. Avoids the axiom of replacement. (Contributed by Scott Fenton, 21-Apr-2011.) (Proof shortened by Scott Fenton, 17-Nov-2024.) |
Ref | Expression |
---|---|
wfrrel.1 | ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) |
Ref | Expression |
---|---|
wfrdmcl | ⊢ (𝑋 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wfrrel.1 | . . 3 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) | |
2 | df-wrecs 8099 | . . 3 ⊢ wrecs(𝑅, 𝐴, 𝐺) = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd )) | |
3 | 1, 2 | eqtri 2766 | . 2 ⊢ 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd )) |
4 | 3 | frrdmcl 8095 | 1 ⊢ (𝑋 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2108 ⊆ wss 3883 dom cdm 5580 ∘ ccom 5584 Predcpred 6190 2nd c2nd 7803 frecscfrecs 8067 wrecscwrecs 8098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2817 df-ral 3068 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-iota 6376 df-fun 6420 df-fn 6421 df-fv 6426 df-ov 7258 df-frecs 8068 df-wrecs 8099 |
This theorem is referenced by: (None) |
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