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Mirrors > Home > ILE Home > Th. List > tposf2 | GIF version |
Description: The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.) |
Ref | Expression |
---|---|
tposf2 | ⊢ (Rel 𝐴 → (𝐹:𝐴⟶𝐵 → tpos 𝐹:◡𝐴⟶𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 5174 | . . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | dffn4 5252 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹) | |
3 | 1, 2 | sylib 121 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→ran 𝐹) |
4 | tposfo2 6046 | . . . . . 6 ⊢ (Rel 𝐴 → (𝐹:𝐴–onto→ran 𝐹 → tpos 𝐹:◡𝐴–onto→ran 𝐹)) | |
5 | 3, 4 | syl5 32 | . . . . 5 ⊢ (Rel 𝐴 → (𝐹:𝐴⟶𝐵 → tpos 𝐹:◡𝐴–onto→ran 𝐹)) |
6 | 5 | imp 123 | . . . 4 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → tpos 𝐹:◡𝐴–onto→ran 𝐹) |
7 | fof 5246 | . . . 4 ⊢ (tpos 𝐹:◡𝐴–onto→ran 𝐹 → tpos 𝐹:◡𝐴⟶ran 𝐹) | |
8 | 6, 7 | syl 14 | . . 3 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → tpos 𝐹:◡𝐴⟶ran 𝐹) |
9 | frn 5182 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
10 | 9 | adantl 272 | . . 3 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → ran 𝐹 ⊆ 𝐵) |
11 | fss 5185 | . . 3 ⊢ ((tpos 𝐹:◡𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → tpos 𝐹:◡𝐴⟶𝐵) | |
12 | 8, 10, 11 | syl2anc 404 | . 2 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → tpos 𝐹:◡𝐴⟶𝐵) |
13 | 12 | ex 114 | 1 ⊢ (Rel 𝐴 → (𝐹:𝐴⟶𝐵 → tpos 𝐹:◡𝐴⟶𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ⊆ wss 3000 ◡ccnv 4450 ran crn 4452 Rel wrel 4456 Fn wfn 5023 ⟶wf 5024 –onto→wfo 5026 tpos ctpos 6023 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-mpt 3907 df-id 4129 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-rn 4462 df-res 4463 df-ima 4464 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-fo 5034 df-fv 5036 df-tpos 6024 |
This theorem is referenced by: tposf 6051 |
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