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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 8243 | . . 3 ⊢ wrecs(𝑅, 𝐴, 𝐺) = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd )) | |
3 | 1, 2 | eqtri 2764 | . 2 ⊢ 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd )) |
4 | 3 | frrdmcl 8239 | 1 ⊢ (𝑋 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ⊆ wss 3910 dom cdm 5633 ∘ ccom 5637 Predcpred 6252 2nd c2nd 7920 frecscfrecs 8211 wrecscwrecs 8242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-iota 6448 df-fun 6498 df-fn 6499 df-fv 6504 df-ov 7360 df-frecs 8212 df-wrecs 8243 |
This theorem is referenced by: (None) |
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