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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 8306 | . 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 4919 | . 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 2775 | 1 ⊢ 𝐹 = ∪ 𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∃wex 1779 {cab 2714 ∀wral 3061 ⊆ wss 3951 ∪ cuni 4907 ↾ cres 5687 Predcpred 6320 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 frecscfrecs 8305 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-uni 4908 df-frecs 8306 |
| This theorem is referenced by: frrlem6 8316 frrlem7 8317 frrlem8 8318 frrlem9 8319 frrlem10 8320 frrlem14 8324 |
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