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Mirrors > Home > ILE Home > Th. List > tfr1onlem3ag | GIF version |
Description: Lemma for transfinite recursion. This lemma changes some bound variables in 𝐴 (version of tfrlem3ag 6286 but for tfr1on 6327 related lemmas). (Contributed by Jim Kingdon, 13-Mar-2022.) |
Ref | Expression |
---|---|
tfr1onlem3ag.1 | ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} |
Ref | Expression |
---|---|
tfr1onlem3ag | ⊢ (𝐻 ∈ 𝑉 → (𝐻 ∈ 𝐴 ↔ ∃𝑧 ∈ 𝑋 (𝐻 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝐻‘𝑤) = (𝐺‘(𝐻 ↾ 𝑤))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fneq12 5289 | . . . 4 ⊢ ((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) → (𝑓 Fn 𝑥 ↔ 𝐻 Fn 𝑧)) | |
2 | simpll 524 | . . . . . . 7 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑓 = 𝐻) | |
3 | simpr 109 | . . . . . . 7 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤) | |
4 | 2, 3 | fveq12d 5501 | . . . . . 6 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓‘𝑦) = (𝐻‘𝑤)) |
5 | 2, 3 | reseq12d 4890 | . . . . . . 7 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓 ↾ 𝑦) = (𝐻 ↾ 𝑤)) |
6 | 5 | fveq2d 5498 | . . . . . 6 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝐺‘(𝑓 ↾ 𝑦)) = (𝐺‘(𝐻 ↾ 𝑤))) |
7 | 4, 6 | eqeq12d 2185 | . . . . 5 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ (𝐻‘𝑤) = (𝐺‘(𝐻 ↾ 𝑤)))) |
8 | simplr 525 | . . . . 5 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧) | |
9 | 7, 8 | cbvraldva2 2703 | . . . 4 ⊢ ((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ ∀𝑤 ∈ 𝑧 (𝐻‘𝑤) = (𝐺‘(𝐻 ↾ 𝑤)))) |
10 | 1, 9 | anbi12d 470 | . . 3 ⊢ ((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) → ((𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) ↔ (𝐻 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝐻‘𝑤) = (𝐺‘(𝐻 ↾ 𝑤))))) |
11 | 10 | cbvrexdva 2706 | . 2 ⊢ (𝑓 = 𝐻 → (∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) ↔ ∃𝑧 ∈ 𝑋 (𝐻 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝐻‘𝑤) = (𝐺‘(𝐻 ↾ 𝑤))))) |
12 | tfr1onlem3ag.1 | . 2 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} | |
13 | 11, 12 | elab2g 2877 | 1 ⊢ (𝐻 ∈ 𝑉 → (𝐻 ∈ 𝐴 ↔ ∃𝑧 ∈ 𝑋 (𝐻 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝐻‘𝑤) = (𝐺‘(𝐻 ↾ 𝑤))))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1348 ∈ wcel 2141 {cab 2156 ∀wral 2448 ∃wrex 2449 ↾ cres 4611 Fn wfn 5191 ‘cfv 5196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-res 4621 df-iota 5158 df-fun 5198 df-fn 5199 df-fv 5204 |
This theorem is referenced by: tfr1onlem3 6315 tfr1onlemsucaccv 6318 tfr1onlembxssdm 6320 tfr1onlemres 6326 |
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