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Mirrors > Home > ILE Home > Th. List > fliftrel | GIF version |
Description: 𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Ref | Expression |
---|---|
flift.1 | ⊢ 𝐹 = ran (x ∈ 𝑋 ↦ 〈A, B〉) |
flift.2 | ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅) |
flift.3 | ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆) |
Ref | Expression |
---|---|
fliftrel | ⊢ (φ → 𝐹 ⊆ (𝑅 × 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flift.1 | . 2 ⊢ 𝐹 = ran (x ∈ 𝑋 ↦ 〈A, B〉) | |
2 | flift.2 | . . . . 5 ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅) | |
3 | flift.3 | . . . . 5 ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆) | |
4 | opelxpi 4319 | . . . . 5 ⊢ ((A ∈ 𝑅 ∧ B ∈ 𝑆) → 〈A, B〉 ∈ (𝑅 × 𝑆)) | |
5 | 2, 3, 4 | syl2anc 391 | . . . 4 ⊢ ((φ ∧ x ∈ 𝑋) → 〈A, B〉 ∈ (𝑅 × 𝑆)) |
6 | eqid 2037 | . . . 4 ⊢ (x ∈ 𝑋 ↦ 〈A, B〉) = (x ∈ 𝑋 ↦ 〈A, B〉) | |
7 | 5, 6 | fmptd 5265 | . . 3 ⊢ (φ → (x ∈ 𝑋 ↦ 〈A, B〉):𝑋⟶(𝑅 × 𝑆)) |
8 | frn 4995 | . . 3 ⊢ ((x ∈ 𝑋 ↦ 〈A, B〉):𝑋⟶(𝑅 × 𝑆) → ran (x ∈ 𝑋 ↦ 〈A, B〉) ⊆ (𝑅 × 𝑆)) | |
9 | 7, 8 | syl 14 | . 2 ⊢ (φ → ran (x ∈ 𝑋 ↦ 〈A, B〉) ⊆ (𝑅 × 𝑆)) |
10 | 1, 9 | syl5eqss 2983 | 1 ⊢ (φ → 𝐹 ⊆ (𝑅 × 𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∈ wcel 1390 ⊆ wss 2911 〈cop 3370 ↦ cmpt 3809 × cxp 4286 ran crn 4289 ⟶wf 4841 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-fv 4853 |
This theorem is referenced by: fliftcnv 5378 fliftfun 5379 fliftf 5382 qliftrel 6121 |
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