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Mirrors > Home > ILE Home > Th. List > tfr1onlemssrecs | GIF version |
Description: Lemma for tfr1on 6215. The union of functions acceptable for tfr1on 6215 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 4378 | . . . . . 6 ⊢ (Ord 𝑋 → 𝑋 ⊆ On) | |
4 | ssrexv 3132 | . . . . . 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 3140 | . . . 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 3113 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}) |
8 | 7 | unissd 3730 | . 2 ⊢ (𝜑 → ∪ 𝐴 ⊆ ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}) |
9 | df-recs 6170 | . 2 ⊢ recs(𝐺) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} | |
10 | 8, 9 | sseqtrrdi 3116 | 1 ⊢ (𝜑 → ∪ 𝐴 ⊆ recs(𝐺)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 {cab 2103 ∀wral 2393 ∃wrex 2394 ⊆ wss 3041 ∪ cuni 3706 Ord word 4254 Oncon0 4255 ↾ cres 4511 Fn wfn 5088 ‘cfv 5093 recscrecs 6169 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-in 3047 df-ss 3054 df-uni 3707 df-tr 3997 df-iord 4258 df-on 4260 df-recs 6170 |
This theorem is referenced by: tfr1onlembfn 6209 tfr1onlemubacc 6211 tfr1onlemres 6214 |
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