Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ctiunctlemu2nd | GIF version |
Description: Lemma for ctiunct 12388. (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 5499 | . . . . . . 7 ⊢ (𝑧 = 𝑁 → (1st ‘(𝐽‘𝑧)) = (1st ‘(𝐽‘𝑁))) | |
3 | 2 | eleq1d 2239 | . . . . . 6 ⊢ (𝑧 = 𝑁 → ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ↔ (1st ‘(𝐽‘𝑁)) ∈ 𝑆)) |
4 | 2fveq3 5499 | . . . . . . 7 ⊢ (𝑧 = 𝑁 → (2nd ‘(𝐽‘𝑧)) = (2nd ‘(𝐽‘𝑁))) | |
5 | 2 | fveq2d 5498 | . . . . . . . 8 ⊢ (𝑧 = 𝑁 → (𝐹‘(1st ‘(𝐽‘𝑧))) = (𝐹‘(1st ‘(𝐽‘𝑁)))) |
6 | 5 | csbeq1d 3056 | . . . . . . 7 ⊢ (𝑧 = 𝑁 → ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇 = ⦋(𝐹‘(1st ‘(𝐽‘𝑁))) / 𝑥⦌𝑇) |
7 | 4, 6 | eleq12d 2241 | . . . . . 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 470 | . . . . 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 2889 | . . . 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 829 = wceq 1348 ∈ wcel 2141 ∀wral 2448 {crab 2452 ⦋csb 3049 ⊆ wss 3121 ωcom 4572 × cxp 4607 –onto→wfo 5194 –1-1-onto→wf1o 5195 ‘cfv 5196 1st c1st 6115 2nd c2nd 6116 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-iota 5158 df-fv 5204 |
This theorem is referenced by: ctiunctlemf 12386 |
Copyright terms: Public domain | W3C validator |