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Mirrors > Home > ILE Home > Th. List > fliftcnv | GIF version |
Description: Converse of the relation 𝐹. (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 |
---|---|
fliftcnv | ⊢ (φ → ◡𝐹 = ran (x ∈ 𝑋 ↦ 〈B, A〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2037 | . . . . 5 ⊢ ran (x ∈ 𝑋 ↦ 〈B, A〉) = ran (x ∈ 𝑋 ↦ 〈B, A〉) | |
2 | flift.3 | . . . . 5 ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆) | |
3 | flift.2 | . . . . 5 ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅) | |
4 | 1, 2, 3 | fliftrel 5375 | . . . 4 ⊢ (φ → ran (x ∈ 𝑋 ↦ 〈B, A〉) ⊆ (𝑆 × 𝑅)) |
5 | relxp 4390 | . . . 4 ⊢ Rel (𝑆 × 𝑅) | |
6 | relss 4370 | . . . 4 ⊢ (ran (x ∈ 𝑋 ↦ 〈B, A〉) ⊆ (𝑆 × 𝑅) → (Rel (𝑆 × 𝑅) → Rel ran (x ∈ 𝑋 ↦ 〈B, A〉))) | |
7 | 4, 5, 6 | mpisyl 1332 | . . 3 ⊢ (φ → Rel ran (x ∈ 𝑋 ↦ 〈B, A〉)) |
8 | relcnv 4646 | . . 3 ⊢ Rel ◡𝐹 | |
9 | 7, 8 | jctil 295 | . 2 ⊢ (φ → (Rel ◡𝐹 ∧ Rel ran (x ∈ 𝑋 ↦ 〈B, A〉))) |
10 | flift.1 | . . . . . . 7 ⊢ 𝐹 = ran (x ∈ 𝑋 ↦ 〈A, B〉) | |
11 | 10, 3, 2 | fliftel 5376 | . . . . . 6 ⊢ (φ → (z𝐹y ↔ ∃x ∈ 𝑋 (z = A ∧ y = B))) |
12 | vex 2554 | . . . . . . 7 ⊢ y ∈ V | |
13 | vex 2554 | . . . . . . 7 ⊢ z ∈ V | |
14 | 12, 13 | brcnv 4461 | . . . . . 6 ⊢ (y◡𝐹z ↔ z𝐹y) |
15 | ancom 253 | . . . . . . 7 ⊢ ((y = B ∧ z = A) ↔ (z = A ∧ y = B)) | |
16 | 15 | rexbii 2325 | . . . . . 6 ⊢ (∃x ∈ 𝑋 (y = B ∧ z = A) ↔ ∃x ∈ 𝑋 (z = A ∧ y = B)) |
17 | 11, 14, 16 | 3bitr4g 212 | . . . . 5 ⊢ (φ → (y◡𝐹z ↔ ∃x ∈ 𝑋 (y = B ∧ z = A))) |
18 | 1, 2, 3 | fliftel 5376 | . . . . 5 ⊢ (φ → (yran (x ∈ 𝑋 ↦ 〈B, A〉)z ↔ ∃x ∈ 𝑋 (y = B ∧ z = A))) |
19 | 17, 18 | bitr4d 180 | . . . 4 ⊢ (φ → (y◡𝐹z ↔ yran (x ∈ 𝑋 ↦ 〈B, A〉)z)) |
20 | df-br 3756 | . . . 4 ⊢ (y◡𝐹z ↔ 〈y, z〉 ∈ ◡𝐹) | |
21 | df-br 3756 | . . . 4 ⊢ (yran (x ∈ 𝑋 ↦ 〈B, A〉)z ↔ 〈y, z〉 ∈ ran (x ∈ 𝑋 ↦ 〈B, A〉)) | |
22 | 19, 20, 21 | 3bitr3g 211 | . . 3 ⊢ (φ → (〈y, z〉 ∈ ◡𝐹 ↔ 〈y, z〉 ∈ ran (x ∈ 𝑋 ↦ 〈B, A〉))) |
23 | 22 | eqrelrdv2 4382 | . 2 ⊢ (((Rel ◡𝐹 ∧ Rel ran (x ∈ 𝑋 ↦ 〈B, A〉)) ∧ φ) → ◡𝐹 = ran (x ∈ 𝑋 ↦ 〈B, A〉)) |
24 | 9, 23 | mpancom 399 | 1 ⊢ (φ → ◡𝐹 = ran (x ∈ 𝑋 ↦ 〈B, A〉)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∈ wcel 1390 ∃wrex 2301 ⊆ wss 2911 〈cop 3370 class class class wbr 3755 ↦ cmpt 3809 × cxp 4286 ◡ccnv 4287 ran crn 4289 Rel wrel 4293 |
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: (None) |
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