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Mirrors > Home > MPE Home > Th. List > tposf2 | Structured version Visualization version GIF version |
Description: The domain and codomain 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 8256 | . . . . 5 ⊢ (Rel 𝐴 → (𝐹:𝐴–onto→ran 𝐹 → tpos 𝐹:◡𝐴–onto→ran 𝐹)) | |
2 | ffn 6720 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
3 | dffn4 6813 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹) | |
4 | 2, 3 | sylib 217 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→ran 𝐹) |
5 | 1, 4 | impel 504 | . . . 4 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → tpos 𝐹:◡𝐴–onto→ran 𝐹) |
6 | fof 6807 | . . . 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 6727 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
9 | 8 | adantl 480 | . . 3 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → ran 𝐹 ⊆ 𝐵) |
10 | 7, 9 | fssd 6737 | . 2 ⊢ ((Rel 𝐴 ∧ 𝐹:𝐴⟶𝐵) → tpos 𝐹:◡𝐴⟶𝐵) |
11 | 10 | ex 411 | 1 ⊢ (Rel 𝐴 → (𝐹:𝐴⟶𝐵 → tpos 𝐹:◡𝐴⟶𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ⊆ wss 3946 ◡ccnv 5673 ran crn 5675 Rel wrel 5679 Fn wfn 6541 ⟶wf 6542 –onto→wfo 6544 tpos ctpos 8232 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-fo 6552 df-fv 6554 df-tpos 8233 |
This theorem is referenced by: tposf 8261 |
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