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Mirrors > Home > MPE Home > Th. List > tposf2 | Structured version Visualization version 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 | tposfo2 7917 | . . . . 5 ⊢ (Rel 𝐴 → (𝐹:𝐴–onto→ran 𝐹 → tpos 𝐹:◡𝐴–onto→ran 𝐹)) | |
2 | ffn 6516 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
3 | dffn4 6598 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹) | |
4 | 2, 3 | sylib 220 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→ran 𝐹) |
5 | 1, 4 | impel 508 | . . . 4 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → tpos 𝐹:◡𝐴–onto→ran 𝐹) |
6 | fof 6592 | . . . 4 ⊢ (tpos 𝐹:◡𝐴–onto→ran 𝐹 → tpos 𝐹:◡𝐴⟶ran 𝐹) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → tpos 𝐹:◡𝐴⟶ran 𝐹) |
8 | frn 6522 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
9 | 8 | adantl 484 | . . 3 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → ran 𝐹 ⊆ 𝐵) |
10 | 7, 9 | fssd 6530 | . 2 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → tpos 𝐹:◡𝐴⟶𝐵) |
11 | 10 | ex 415 | 1 ⊢ (Rel 𝐴 → (𝐹:𝐴⟶𝐵 → tpos 𝐹:◡𝐴⟶𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ⊆ wss 3938 ◡ccnv 5556 ran crn 5558 Rel wrel 5562 Fn wfn 6352 ⟶wf 6353 –onto→wfo 6355 tpos ctpos 7893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fo 6363 df-fv 6365 df-tpos 7894 |
This theorem is referenced by: tposf 7922 |
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