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| Mirrors > Home > MPE Home > Th. List > wfrdmss | Structured version Visualization version GIF version | ||
| Description: The domain of the well-ordered recursion generator is a subclass of 𝐴. 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 | 
|---|---|
| wfrdmss | ⊢ dom 𝐹 ⊆ 𝐴 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | wfrrel.1 | . . 3 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) | |
| 2 | df-wrecs 8338 | . . 3 ⊢ wrecs(𝑅, 𝐴, 𝐺) = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd )) | |
| 3 | 1, 2 | eqtri 2764 | . 2 ⊢ 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd )) | 
| 4 | 3 | frrdmss 8333 | 1 ⊢ dom 𝐹 ⊆ 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ⊆ wss 3950 dom cdm 5684 ∘ ccom 5688 2nd c2nd 8014 frecscfrecs 8306 wrecscwrecs 8337 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-iota 6513 df-fun 6562 df-fn 6563 df-fv 6568 df-ov 7435 df-frecs 8307 df-wrecs 8338 | 
| This theorem is referenced by: (None) | 
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