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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 8244 | . . 3 ⊢ wrecs(𝑅, 𝐴, 𝐺) = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd )) | |
3 | 1, 2 | eqtri 2761 | . 2 ⊢ 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd )) |
4 | 3 | frrdmss 8239 | 1 ⊢ dom 𝐹 ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ⊆ wss 3911 dom cdm 5634 ∘ ccom 5638 2nd c2nd 7921 frecscfrecs 8212 wrecscwrecs 8243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-iota 6449 df-fun 6499 df-fn 6500 df-fv 6505 df-ov 7361 df-frecs 8213 df-wrecs 8244 |
This theorem is referenced by: (None) |
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