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Mirrors > Home > ILE Home > Th. List > ctiunctlemuom | GIF version |
Description: Lemma for ctiunct 12441. (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 3241 | . . 3 ⊢ {𝑧 ∈ ω ∣ ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)} ⊆ ω | |
3 | 1, 2 | eqsstri 3188 | . 2 ⊢ 𝑈 ⊆ ω |
4 | 3 | a1i 9 | 1 ⊢ (𝜑 → 𝑈 ⊆ ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 DECID wdc 834 = wceq 1353 ∈ wcel 2148 ∀wral 2455 {crab 2459 ⦋csb 3058 ⊆ wss 3130 ωcom 4590 × cxp 4625 –onto→wfo 5215 –1-1-onto→wf1o 5216 ‘cfv 5217 1st c1st 6139 2nd c2nd 6140 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rab 2464 df-in 3136 df-ss 3143 |
This theorem is referenced by: ctiunct 12441 |
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