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Mirrors > Home > MPE Home > Th. List > frrlem6 | Structured version Visualization version GIF version |
Description: Lemma for well-founded recursion. The well-founded recursion generator is a relation. (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 |
---|---|
frrlem6 | ⊢ Rel 𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frrlem5.1 | . . . . 5 ⊢ 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))))} | |
2 | frrlem5.2 | . . . . 5 ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) | |
3 | 1, 2 | frrlem5 8277 | . . . 4 ⊢ 𝐹 = ∪ 𝐵 |
4 | 3 | releqi 5777 | . . 3 ⊢ (Rel 𝐹 ↔ Rel ∪ 𝐵) |
5 | reluni 5818 | . . 3 ⊢ (Rel ∪ 𝐵 ↔ ∀𝑔 ∈ 𝐵 Rel 𝑔) | |
6 | 4, 5 | bitri 274 | . 2 ⊢ (Rel 𝐹 ↔ ∀𝑔 ∈ 𝐵 Rel 𝑔) |
7 | 1 | frrlem2 8274 | . . 3 ⊢ (𝑔 ∈ 𝐵 → Fun 𝑔) |
8 | funrel 6565 | . . 3 ⊢ (Fun 𝑔 → Rel 𝑔) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝑔 ∈ 𝐵 → Rel 𝑔) |
10 | 6, 9 | mprgbir 3068 | 1 ⊢ Rel 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 {cab 2709 ∀wral 3061 ⊆ wss 3948 ∪ cuni 4908 ↾ cres 5678 Rel wrel 5681 Predcpred 6299 Fun wfun 6537 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-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-iota 6495 df-fun 6545 df-fn 6546 df-fv 6551 df-ov 7414 df-frecs 8268 |
This theorem is referenced by: frrlem9 8281 frrrel 8293 |
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