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Mirrors > Home > ILE Home > Th. List > ctiunctlemu2nd | 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)} |
ctiunctlem.n | ⊢ (𝜑 → 𝑁 ∈ 𝑈) |
Ref | Expression |
---|---|
ctiunctlemu2nd | ⊢ (𝜑 → (2nd ‘(𝐽‘𝑁)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑁))) / 𝑥⦌𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ctiunctlem.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑈) | |
2 | 2fveq3 5434 | . . . . . . 7 ⊢ (𝑧 = 𝑁 → (1st ‘(𝐽‘𝑧)) = (1st ‘(𝐽‘𝑁))) | |
3 | 2 | eleq1d 2209 | . . . . . 6 ⊢ (𝑧 = 𝑁 → ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ↔ (1st ‘(𝐽‘𝑁)) ∈ 𝑆)) |
4 | 2fveq3 5434 | . . . . . . 7 ⊢ (𝑧 = 𝑁 → (2nd ‘(𝐽‘𝑧)) = (2nd ‘(𝐽‘𝑁))) | |
5 | 2 | fveq2d 5433 | . . . . . . . 8 ⊢ (𝑧 = 𝑁 → (𝐹‘(1st ‘(𝐽‘𝑧))) = (𝐹‘(1st ‘(𝐽‘𝑁)))) |
6 | 5 | csbeq1d 3014 | . . . . . . 7 ⊢ (𝑧 = 𝑁 → ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇 = ⦋(𝐹‘(1st ‘(𝐽‘𝑁))) / 𝑥⦌𝑇) |
7 | 4, 6 | eleq12d 2211 | . . . . . 6 ⊢ (𝑧 = 𝑁 → ((2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇 ↔ (2nd ‘(𝐽‘𝑁)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑁))) / 𝑥⦌𝑇)) |
8 | 3, 7 | anbi12d 465 | . . . . 5 ⊢ (𝑧 = 𝑁 → (((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇) ↔ ((1st ‘(𝐽‘𝑁)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑁)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑁))) / 𝑥⦌𝑇))) |
9 | ctiunct.u | . . . . 5 ⊢ 𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)} | |
10 | 8, 9 | elrab2 2847 | . . . 4 ⊢ (𝑁 ∈ 𝑈 ↔ (𝑁 ∈ ω ∧ ((1st ‘(𝐽‘𝑁)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑁)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑁))) / 𝑥⦌𝑇))) |
11 | 1, 10 | sylib 121 | . . 3 ⊢ (𝜑 → (𝑁 ∈ ω ∧ ((1st ‘(𝐽‘𝑁)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑁)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑁))) / 𝑥⦌𝑇))) |
12 | 11 | simprd 113 | . 2 ⊢ (𝜑 → ((1st ‘(𝐽‘𝑁)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑁)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑁))) / 𝑥⦌𝑇)) |
13 | 12 | simprd 113 | 1 ⊢ (𝜑 → (2nd ‘(𝐽‘𝑁)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑁))) / 𝑥⦌𝑇) |
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-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-un 3080 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-iota 5096 df-fv 5139 |
This theorem is referenced by: ctiunctlemf 11987 |
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