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Mirrors > Home > ILE Home > Th. List > tposf1o2 | GIF version |
Description: Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.) |
Ref | Expression |
---|---|
tposf1o2 | ⊢ (Rel A → (𝐹:A–1-1-onto→B → tpos 𝐹:◡A–1-1-onto→B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tposf12 5825 | . . 3 ⊢ (Rel A → (𝐹:A–1-1→B → tpos 𝐹:◡A–1-1→B)) | |
2 | tposfo2 5823 | . . 3 ⊢ (Rel A → (𝐹:A–onto→B → tpos 𝐹:◡A–onto→B)) | |
3 | 1, 2 | anim12d 318 | . 2 ⊢ (Rel A → ((𝐹:A–1-1→B ∧ 𝐹:A–onto→B) → (tpos 𝐹:◡A–1-1→B ∧ tpos 𝐹:◡A–onto→B))) |
4 | df-f1o 4852 | . 2 ⊢ (𝐹:A–1-1-onto→B ↔ (𝐹:A–1-1→B ∧ 𝐹:A–onto→B)) | |
5 | df-f1o 4852 | . 2 ⊢ (tpos 𝐹:◡A–1-1-onto→B ↔ (tpos 𝐹:◡A–1-1→B ∧ tpos 𝐹:◡A–onto→B)) | |
6 | 3, 4, 5 | 3imtr4g 194 | 1 ⊢ (Rel A → (𝐹:A–1-1-onto→B → tpos 𝐹:◡A–1-1-onto→B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ◡ccnv 4287 Rel wrel 4293 –1-1→wf1 4842 –onto→wfo 4843 –1-1-onto→wf1o 4844 tpos ctpos 5800 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-sbc 2759 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-1st 5709 df-2nd 5710 df-tpos 5801 |
This theorem is referenced by: (None) |
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