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Mirrors > Home > MPE Home > Th. List > frrlem5 | Structured version Visualization version GIF version |
Description: Lemma for well-founded recursion. State the well-founded recursion generator in terms of the acceptable functions. (Contributed by Scott Fenton, 27-Aug-2022.) |
Ref | Expression |
---|---|
frrlem5.1 | ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} |
frrlem5.2 | ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) |
Ref | Expression |
---|---|
frrlem5 | ⊢ 𝐹 = ∪ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-frecs 8268 | . 2 ⊢ frecs(𝑅, 𝐴, 𝐺) = ∪ {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 | frrlem5.2 | . 2 ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) | |
3 | frrlem5.1 | . . 3 ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} | |
4 | 3 | unieqi 4921 | . 2 ⊢ ∪ 𝐵 = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} |
5 | 1, 2, 4 | 3eqtr4i 2770 | 1 ⊢ 𝐹 = ∪ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 {cab 2709 ∀wral 3061 ⊆ wss 3948 ∪ cuni 4908 ↾ cres 5678 Predcpred 6299 Fn wfn 6538 ‘cfv 6543 (class class class)co 7411 frecscfrecs 8267 |
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-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-in 3955 df-ss 3965 df-uni 4909 df-frecs 8268 |
This theorem is referenced by: frrlem6 8278 frrlem7 8279 frrlem8 8280 frrlem9 8281 frrlem10 8282 frrlem14 8286 |
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