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Mirrors > Home > ILE Home > Th. List > tposfn2 | 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 6236 | . . . 4 ⊢ (Fun 𝐹 → Fun tpos 𝐹) | |
2 | 1 | a1i 9 | . . 3 ⊢ (Rel 𝐴 → (Fun 𝐹 → Fun tpos 𝐹)) |
3 | dmtpos 6232 | . . . . . 6 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹) | |
4 | 3 | a1i 9 | . . . . 5 ⊢ (dom 𝐹 = 𝐴 → (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)) |
5 | releq 4691 | . . . . 5 ⊢ (dom 𝐹 = 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴)) | |
6 | cnveq 4783 | . . . . . 6 ⊢ (dom 𝐹 = 𝐴 → ◡dom 𝐹 = ◡𝐴) | |
7 | 6 | eqeq2d 2182 | . . . . 5 ⊢ (dom 𝐹 = 𝐴 → (dom tpos 𝐹 = ◡dom 𝐹 ↔ dom tpos 𝐹 = ◡𝐴)) |
8 | 4, 5, 7 | 3imtr3d 201 | . . . 4 ⊢ (dom 𝐹 = 𝐴 → (Rel 𝐴 → dom tpos 𝐹 = ◡𝐴)) |
9 | 8 | com12 30 | . . 3 ⊢ (Rel 𝐴 → (dom 𝐹 = 𝐴 → dom tpos 𝐹 = ◡𝐴)) |
10 | 2, 9 | anim12d 333 | . 2 ⊢ (Rel 𝐴 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (Fun tpos 𝐹 ∧ dom tpos 𝐹 = ◡𝐴))) |
11 | df-fn 5199 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
12 | df-fn 5199 | . 2 ⊢ (tpos 𝐹 Fn ◡𝐴 ↔ (Fun tpos 𝐹 ∧ dom tpos 𝐹 = ◡𝐴)) | |
13 | 10, 11, 12 | 3imtr4g 204 | 1 ⊢ (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn ◡𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ◡ccnv 4608 dom cdm 4609 Rel wrel 4614 Fun wfun 5190 Fn wfn 5191 tpos ctpos 6220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-fv 5204 df-tpos 6221 |
This theorem is referenced by: tposfo2 6243 tpos0 6250 |
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