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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 8239 | . . . 4 ⊢ (Fun 𝐹 → Fun tpos 𝐹) | |
2 | 1 | a1i 11 | . . 3 ⊢ (Rel 𝐴 → (Fun 𝐹 → Fun tpos 𝐹)) |
3 | dmtpos 8235 | . . . . . 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 5772 | . . . . 5 ⊢ (dom 𝐹 = 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴)) | |
6 | cnveq 5870 | . . . . . 6 ⊢ (dom 𝐹 = 𝐴 → ◡dom 𝐹 = ◡𝐴) | |
7 | 6 | eqeq2d 2738 | . . . . 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 608 | . 2 ⊢ (Rel 𝐴 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (Fun tpos 𝐹 ∧ dom tpos 𝐹 = ◡𝐴))) |
11 | df-fn 6545 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
12 | df-fn 6545 | . 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 1534 ◡ccnv 5671 dom cdm 5672 Rel wrel 5677 Fun wfun 6536 Fn wfn 6537 tpos ctpos 8222 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-fv 6550 df-tpos 8223 |
This theorem is referenced by: tposfo2 8246 tpos0 8253 |
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