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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 8275 | . . . 4 ⊢ 𝐹 = ∪ 𝐵 |
4 | 3 | releqi 5778 | . . 3 ⊢ (Rel 𝐹 ↔ Rel ∪ 𝐵) |
5 | reluni 5819 | . . 3 ⊢ (Rel ∪ 𝐵 ↔ ∀𝑔 ∈ 𝐵 Rel 𝑔) | |
6 | 4, 5 | bitri 275 | . 2 ⊢ (Rel 𝐹 ↔ ∀𝑔 ∈ 𝐵 Rel 𝑔) |
7 | 1 | frrlem2 8272 | . . 3 ⊢ (𝑔 ∈ 𝐵 → Fun 𝑔) |
8 | funrel 6566 | . . 3 ⊢ (Fun 𝑔 → Rel 𝑔) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝑔 ∈ 𝐵 → Rel 𝑔) |
10 | 6, 9 | mprgbir 3069 | 1 ⊢ Rel 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2710 ∀wral 3062 ⊆ wss 3949 ∪ cuni 4909 ↾ cres 5679 Rel wrel 5682 Predcpred 6300 Fun wfun 6538 Fn wfn 6539 ‘cfv 6544 (class class class)co 7409 frecscfrecs 8265 |
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 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-iota 6496 df-fun 6546 df-fn 6547 df-fv 6552 df-ov 7412 df-frecs 8266 |
This theorem is referenced by: frrlem9 8279 frrrel 8291 |
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