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Mirrors > Home > ILE Home > Th. List > ctiunctlemuom | GIF version |
Description: Lemma for ctiunct 11989. (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 3187 | . . 3 ⊢ {𝑧 ∈ ω ∣ ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)} ⊆ ω | |
3 | 1, 2 | eqsstri 3134 | . 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 820 = wceq 1332 ∈ wcel 1481 ∀wral 2417 {crab 2421 ⦋csb 3007 ⊆ wss 3076 ωcom 4512 × cxp 4545 –onto→wfo 5129 –1-1-onto→wf1o 5130 ‘cfv 5131 1st c1st 6044 2nd c2nd 6045 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rab 2426 df-in 3082 df-ss 3089 |
This theorem is referenced by: ctiunct 11989 |
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