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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 5117 | . . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | dffn4 5190 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹) | |
3 | 1, 2 | sylib 120 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→ran 𝐹) |
4 | tposfo2 5967 | . . . . . 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 122 | . . . 4 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → tpos 𝐹:◡𝐴–onto→ran 𝐹) |
7 | fof 5184 | . . . 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 5124 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
10 | 9 | adantl 271 | . . 3 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → ran 𝐹 ⊆ 𝐵) |
11 | fss 5126 | . . 3 ⊢ ((tpos 𝐹:◡𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → tpos 𝐹:◡𝐴⟶𝐵) | |
12 | 8, 10, 11 | syl2anc 403 | . 2 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → tpos 𝐹:◡𝐴⟶𝐵) |
13 | 12 | ex 113 | 1 ⊢ (Rel 𝐴 → (𝐹:𝐴⟶𝐵 → tpos 𝐹:◡𝐴⟶𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ⊆ wss 2986 ◡ccnv 4403 ran crn 4405 Rel wrel 4409 Fn wfn 4967 ⟶wf 4968 –onto→wfo 4970 tpos ctpos 5944 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3925 ax-nul 3933 ax-pow 3977 ax-pr 4003 ax-un 4227 |
This theorem depends on definitions: df-bi 115 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-ral 2360 df-rex 2361 df-rab 2364 df-v 2616 df-sbc 2829 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-nul 3273 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-br 3815 df-opab 3869 df-mpt 3870 df-id 4087 df-xp 4410 df-rel 4411 df-cnv 4412 df-co 4413 df-dm 4414 df-rn 4415 df-res 4416 df-ima 4417 df-iota 4937 df-fun 4974 df-fn 4975 df-f 4976 df-fo 4978 df-fv 4980 df-tpos 5945 |
This theorem is referenced by: tposf 5972 |
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