![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > wfrlem3OLDa | Structured version Visualization version GIF version |
Description: Obsolete version as of 18-Nov-2024. Lemma for well-ordered recursion. Show membership in the class of acceptable functions. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 31-Jul-2020.) |
Ref | Expression |
---|---|
wfrlem1OLD.1 | ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} |
wfrlem3OLDa.2 | ⊢ 𝐺 ∈ V |
Ref | Expression |
---|---|
wfrlem3OLDa | ⊢ (𝐺 ∈ 𝐵 ↔ ∃𝑧(𝐺 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wfrlem3OLDa.2 | . 2 ⊢ 𝐺 ∈ V | |
2 | fneq1 6660 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑔 Fn 𝑧 ↔ 𝐺 Fn 𝑧)) | |
3 | fveq1 6906 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑤) = (𝐺‘𝑤)) | |
4 | reseq1 5994 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑤))) | |
5 | 4 | fveq2d 6911 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤)))) |
6 | 3, 5 | eqeq12d 2751 | . . . . 5 ⊢ (𝑔 = 𝐺 → ((𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))) ↔ (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤))))) |
7 | 6 | ralbidv 3176 | . . . 4 ⊢ (𝑔 = 𝐺 → (∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))) ↔ ∀𝑤 ∈ 𝑧 (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤))))) |
8 | 2, 7 | 3anbi13d 1437 | . . 3 ⊢ (𝑔 = 𝐺 → ((𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) ↔ (𝐺 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤)))))) |
9 | 8 | exbidv 1919 | . 2 ⊢ (𝑔 = 𝐺 → (∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) ↔ ∃𝑧(𝐺 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤)))))) |
10 | wfrlem1OLD.1 | . . 3 ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} | |
11 | 10 | wfrlem1OLD 8347 | . 2 ⊢ 𝐵 = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))} |
12 | 1, 9, 11 | elab2 3685 | 1 ⊢ (𝐺 ∈ 𝐵 ↔ ∃𝑧(𝐺 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤))))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∃wex 1776 ∈ wcel 2106 {cab 2712 ∀wral 3059 Vcvv 3478 ⊆ wss 3963 ↾ cres 5691 Predcpred 6322 Fn wfn 6558 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 |
This theorem is referenced by: wfrlem17OLD 8364 |
Copyright terms: Public domain | W3C validator |