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Mirrors > Home > ILE Home > Th. List > tposfo2 | GIF version |
Description: Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.) |
Ref | Expression |
---|---|
tposfo2 | ⊢ (Rel 𝐴 → (𝐹:𝐴–onto→𝐵 → tpos 𝐹:◡𝐴–onto→𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tposfn2 6115 | . . . 4 ⊢ (Rel 𝐴 → (𝐹 Fn 𝐴 → tpos 𝐹 Fn ◡𝐴)) | |
2 | 1 | adantrd 275 | . . 3 ⊢ (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → tpos 𝐹 Fn ◡𝐴)) |
3 | fndm 5178 | . . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
4 | 3 | releqd 4581 | . . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → (Rel dom 𝐹 ↔ Rel 𝐴)) |
5 | 4 | biimparc 295 | . . . . . . 7 ⊢ ((Rel 𝐴 ∧ 𝐹 Fn 𝐴) → Rel dom 𝐹) |
6 | rntpos 6106 | . . . . . . 7 ⊢ (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹) | |
7 | 5, 6 | syl 14 | . . . . . 6 ⊢ ((Rel 𝐴 ∧ 𝐹 Fn 𝐴) → ran tpos 𝐹 = ran 𝐹) |
8 | 7 | eqeq1d 2121 | . . . . 5 ⊢ ((Rel 𝐴 ∧ 𝐹 Fn 𝐴) → (ran tpos 𝐹 = 𝐵 ↔ ran 𝐹 = 𝐵)) |
9 | 8 | biimprd 157 | . . . 4 ⊢ ((Rel 𝐴 ∧ 𝐹 Fn 𝐴) → (ran 𝐹 = 𝐵 → ran tpos 𝐹 = 𝐵)) |
10 | 9 | expimpd 358 | . . 3 ⊢ (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → ran tpos 𝐹 = 𝐵)) |
11 | 2, 10 | jcad 303 | . 2 ⊢ (Rel 𝐴 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → (tpos 𝐹 Fn ◡𝐴 ∧ ran tpos 𝐹 = 𝐵))) |
12 | df-fo 5085 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) | |
13 | df-fo 5085 | . 2 ⊢ (tpos 𝐹:◡𝐴–onto→𝐵 ↔ (tpos 𝐹 Fn ◡𝐴 ∧ ran tpos 𝐹 = 𝐵)) | |
14 | 11, 12, 13 | 3imtr4g 204 | 1 ⊢ (Rel 𝐴 → (𝐹:𝐴–onto→𝐵 → tpos 𝐹:◡𝐴–onto→𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1312 ◡ccnv 4496 dom cdm 4497 ran crn 4498 Rel wrel 4502 Fn wfn 5074 –onto→wfo 5077 tpos ctpos 6093 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 ax-un 4313 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-ral 2393 df-rex 2394 df-rab 2397 df-v 2657 df-sbc 2877 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-fo 5085 df-fv 5087 df-tpos 6094 |
This theorem is referenced by: tposf2 6117 tposf1o2 6119 tposfo 6120 |
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