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Mirrors > Home > MPE Home > Th. List > frrdmss | Structured version Visualization version GIF version |
Description: Show without using the axiom of replacement that the domain of the well-founded recursion generator is a subclass of 𝐴. (Contributed by Scott Fenton, 18-Nov-2024.) |
Ref | Expression |
---|---|
frrrel.1 | ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) |
Ref | Expression |
---|---|
frrdmss | ⊢ dom 𝐹 ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2727 | . 2 ⊢ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} | |
2 | frrrel.1 | . 2 ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) | |
3 | 1, 2 | frrlem7 8302 | 1 ⊢ dom 𝐹 ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∃wex 1773 {cab 2704 ∀wral 3057 ⊆ wss 3947 dom cdm 5680 ↾ cres 5682 Predcpred 6307 Fn wfn 6546 ‘cfv 6551 (class class class)co 7424 frecscfrecs 8290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-iota 6503 df-fun 6553 df-fn 6554 df-fv 6559 df-ov 7427 df-frecs 8291 |
This theorem is referenced by: wfrdmss 8355 |
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