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Mirrors > Home > ILE Home > Th. List > ctiunctlemuom | GIF version |
Description: Lemma for ctiunct 11958. (Contributed by Jim Kingdon, 28-Oct-2023.) |
Ref | Expression |
---|---|
ctiunct.som | ⊢ (𝜑 → 𝑆 ⊆ ω) |
ctiunct.sdc | ⊢ (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆) |
ctiunct.f | ⊢ (𝜑 → 𝐹:𝑆–onto→𝐴) |
ctiunct.tom | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 ⊆ ω) |
ctiunct.tdc | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇) |
ctiunct.g | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:𝑇–onto→𝐵) |
ctiunct.j | ⊢ (𝜑 → 𝐽:ω–1-1-onto→(ω × ω)) |
ctiunct.u | ⊢ 𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)} |
Ref | Expression |
---|---|
ctiunctlemuom | ⊢ (𝜑 → 𝑈 ⊆ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ctiunct.u | . . 3 ⊢ 𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)} | |
2 | ssrab2 3182 | . . 3 ⊢ {𝑧 ∈ ω ∣ ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)} ⊆ ω | |
3 | 1, 2 | eqsstri 3129 | . 2 ⊢ 𝑈 ⊆ ω |
4 | 3 | a1i 9 | 1 ⊢ (𝜑 → 𝑈 ⊆ ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 DECID wdc 819 = wceq 1331 ∈ wcel 1480 ∀wral 2416 {crab 2420 ⦋csb 3003 ⊆ wss 3071 ωcom 4504 × cxp 4537 –onto→wfo 5121 –1-1-onto→wf1o 5122 ‘cfv 5123 1st c1st 6036 2nd c2nd 6037 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rab 2425 df-in 3077 df-ss 3084 |
This theorem is referenced by: ctiunct 11958 |
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