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| Mirrors > Home > ILE Home > Th. List > tfr1onlemssrecs | GIF version | ||
| Description: Lemma for tfr1on 6435. The union of functions acceptable for tfr1on 6435 is a subset of recs. (Contributed by Jim Kingdon, 15-Mar-2022.) |
| Ref | Expression |
|---|---|
| tfr1onlemssrecs.1 | ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} |
| tfr1onlemssrecs.x | ⊢ (𝜑 → Ord 𝑋) |
| Ref | Expression |
|---|---|
| tfr1onlemssrecs | ⊢ (𝜑 → ∪ 𝐴 ⊆ recs(𝐺)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tfr1onlemssrecs.1 | . . . 4 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} | |
| 2 | tfr1onlemssrecs.x | . . . . . 6 ⊢ (𝜑 → Ord 𝑋) | |
| 3 | ordsson 4539 | . . . . . 6 ⊢ (Ord 𝑋 → 𝑋 ⊆ On) | |
| 4 | ssrexv 3257 | . . . . . 6 ⊢ (𝑋 ⊆ On → (∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) → ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))))) | |
| 5 | 2, 3, 4 | 3syl 17 | . . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) → ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))))) |
| 6 | 5 | ss2abdv 3265 | . . . 4 ⊢ (𝜑 → {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} ⊆ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}) |
| 7 | 1, 6 | eqsstrid 3238 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}) |
| 8 | 7 | unissd 3873 | . 2 ⊢ (𝜑 → ∪ 𝐴 ⊆ ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}) |
| 9 | df-recs 6390 | . 2 ⊢ recs(𝐺) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} | |
| 10 | 8, 9 | sseqtrrdi 3241 | 1 ⊢ (𝜑 → ∪ 𝐴 ⊆ recs(𝐺)) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∧ wa 104 = wceq 1372 {cab 2190 ∀wral 2483 ∃wrex 2484 ⊆ wss 3165 ∪ cuni 3849 Ord word 4408 Oncon0 4409 ↾ cres 4676 Fn wfn 5265 ‘cfv 5270 recscrecs 6389 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-ext 2186 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-nf 1483 df-sb 1785 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ral 2488 df-rex 2489 df-v 2773 df-in 3171 df-ss 3178 df-uni 3850 df-tr 4142 df-iord 4412 df-on 4414 df-recs 6390 |
| This theorem is referenced by: tfr1onlembfn 6429 tfr1onlemubacc 6431 tfr1onlemres 6434 |
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