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| 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 8316 | . . . 4 ⊢ 𝐹 = ∪ 𝐵 | 
| 4 | 3 | releqi 5786 | . . 3 ⊢ (Rel 𝐹 ↔ Rel ∪ 𝐵) | 
| 5 | reluni 5827 | . . 3 ⊢ (Rel ∪ 𝐵 ↔ ∀𝑔 ∈ 𝐵 Rel 𝑔) | |
| 6 | 4, 5 | bitri 275 | . 2 ⊢ (Rel 𝐹 ↔ ∀𝑔 ∈ 𝐵 Rel 𝑔) | 
| 7 | 1 | frrlem2 8313 | . . 3 ⊢ (𝑔 ∈ 𝐵 → Fun 𝑔) | 
| 8 | funrel 6582 | . . 3 ⊢ (Fun 𝑔 → Rel 𝑔) | |
| 9 | 7, 8 | syl 17 | . 2 ⊢ (𝑔 ∈ 𝐵 → Rel 𝑔) | 
| 10 | 6, 9 | mprgbir 3067 | 1 ⊢ Rel 𝐹 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∃wex 1778 ∈ wcel 2107 {cab 2713 ∀wral 3060 ⊆ wss 3950 ∪ cuni 4906 ↾ cres 5686 Rel wrel 5689 Predcpred 6319 Fun wfun 6554 Fn wfn 6555 ‘cfv 6560 (class class class)co 7432 frecscfrecs 8306 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-iota 6513 df-fun 6562 df-fn 6563 df-fv 6568 df-ov 7435 df-frecs 8307 | 
| This theorem is referenced by: frrlem9 8320 frrrel 8332 | 
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