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| 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 6659 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑔 Fn 𝑧 ↔ 𝐺 Fn 𝑧)) | |
| 3 | fveq1 6905 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑤) = (𝐺‘𝑤)) | |
| 4 | reseq1 5991 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑤))) | |
| 5 | 4 | fveq2d 6910 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤)))) |
| 6 | 3, 5 | eqeq12d 2753 | . . . . 5 ⊢ (𝑔 = 𝐺 → ((𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))) ↔ (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤))))) |
| 7 | 6 | ralbidv 3178 | . . . 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 1440 | . . 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 1921 | . 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 8348 | . 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 3682 | 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 1087 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 ↾ cres 5687 Predcpred 6320 Fn wfn 6556 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 |
| This theorem is referenced by: wfrlem17OLD 8365 |
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