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Mirrors > Home > MPE Home > Th. List > wfrlem3OLDa | Structured version Visualization version GIF version |
Description: Lemma for well-ordered recursion. Show membership in the class of acceptable functions. Obsolete as of 18-Nov-2024. (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 6639 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑔 Fn 𝑧 ↔ 𝐺 Fn 𝑧)) | |
3 | fveq1 6890 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑤) = (𝐺‘𝑤)) | |
4 | reseq1 5973 | . . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑤))) | |
5 | 4 | fveq2d 6895 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤)))) |
6 | 3, 5 | eqeq12d 2743 | . . . . 5 ⊢ (𝑔 = 𝐺 → ((𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))) ↔ (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤))))) |
7 | 6 | ralbidv 3172 | . . . 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 1435 | . . 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 1917 | . 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 8320 | . 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 3669 | 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 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∃wex 1774 ∈ wcel 2099 {cab 2704 ∀wral 3056 Vcvv 3469 ⊆ wss 3944 ↾ cres 5674 Predcpred 6298 Fn wfn 6537 ‘cfv 6542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-iota 6494 df-fun 6544 df-fn 6545 df-fv 6550 |
This theorem is referenced by: wfrlem17OLD 8337 |
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