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| Mirrors > Home > MPE Home > Th. List > tposfn2 | Structured version Visualization version GIF version | ||
| Description: The domain of a transposition. (Contributed by NM, 10-Sep-2015.) |
| Ref | Expression |
|---|---|
| tposfn2 | ⊢ (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn ◡𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tposfun 8267 | . . . 4 ⊢ (Fun 𝐹 → Fun tpos 𝐹) | |
| 2 | 1 | a1i 11 | . . 3 ⊢ (Rel 𝐴 → (Fun 𝐹 → Fun tpos 𝐹)) |
| 3 | dmtpos 8263 | . . . . . 6 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹) | |
| 4 | 3 | a1i 11 | . . . . 5 ⊢ (dom 𝐹 = 𝐴 → (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)) |
| 5 | releq 5786 | . . . . 5 ⊢ (dom 𝐹 = 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴)) | |
| 6 | cnveq 5884 | . . . . . 6 ⊢ (dom 𝐹 = 𝐴 → ◡dom 𝐹 = ◡𝐴) | |
| 7 | 6 | eqeq2d 2748 | . . . . 5 ⊢ (dom 𝐹 = 𝐴 → (dom tpos 𝐹 = ◡dom 𝐹 ↔ dom tpos 𝐹 = ◡𝐴)) |
| 8 | 4, 5, 7 | 3imtr3d 293 | . . . 4 ⊢ (dom 𝐹 = 𝐴 → (Rel 𝐴 → dom tpos 𝐹 = ◡𝐴)) |
| 9 | 8 | com12 32 | . . 3 ⊢ (Rel 𝐴 → (dom 𝐹 = 𝐴 → dom tpos 𝐹 = ◡𝐴)) |
| 10 | 2, 9 | anim12d 609 | . 2 ⊢ (Rel 𝐴 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (Fun tpos 𝐹 ∧ dom tpos 𝐹 = ◡𝐴))) |
| 11 | df-fn 6564 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
| 12 | df-fn 6564 | . 2 ⊢ (tpos 𝐹 Fn ◡𝐴 ↔ (Fun tpos 𝐹 ∧ dom tpos 𝐹 = ◡𝐴)) | |
| 13 | 10, 11, 12 | 3imtr4g 296 | 1 ⊢ (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn ◡𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ◡ccnv 5684 dom cdm 5685 Rel wrel 5690 Fun wfun 6555 Fn wfn 6556 tpos ctpos 8250 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-tpos 8251 |
| This theorem is referenced by: tposfo2 8274 tpos0 8281 tposideq 48788 |
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