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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 relationship. (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 8009 | . . . 4 ⊢ 𝐹 = ∪ 𝐵 |
4 | 3 | releqi 5634 | . . 3 ⊢ (Rel 𝐹 ↔ Rel ∪ 𝐵) |
5 | reluni 5673 | . . 3 ⊢ (Rel ∪ 𝐵 ↔ ∀𝑔 ∈ 𝐵 Rel 𝑔) | |
6 | 4, 5 | bitri 278 | . 2 ⊢ (Rel 𝐹 ↔ ∀𝑔 ∈ 𝐵 Rel 𝑔) |
7 | 1 | frrlem2 8006 | . . 3 ⊢ (𝑔 ∈ 𝐵 → Fun 𝑔) |
8 | funrel 6375 | . . 3 ⊢ (Fun 𝑔 → Rel 𝑔) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝑔 ∈ 𝐵 → Rel 𝑔) |
10 | 6, 9 | mprgbir 3066 | 1 ⊢ Rel 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∃wex 1787 ∈ wcel 2112 {cab 2714 ∀wral 3051 ⊆ wss 3853 ∪ cuni 4805 ↾ cres 5538 Rel wrel 5541 Predcpred 6139 Fun wfun 6352 Fn wfn 6353 ‘cfv 6358 (class class class)co 7191 frecscfrecs 8000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-iota 6316 df-fun 6360 df-fn 6361 df-fv 6366 df-ov 7194 df-frecs 8001 |
This theorem is referenced by: frrlem9 8013 |
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