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Mirrors > Home > MPE Home > Th. List > f1ocnvb | Structured version Visualization version GIF version |
Description: A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and range interchanged. (Contributed by NM, 8-Dec-2003.) |
Ref | Expression |
---|---|
f1ocnvb | ⊢ (Rel 𝐹 → (𝐹:𝐴–1-1-onto→𝐵 ↔ ◡𝐹:𝐵–1-1-onto→𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ocnv 6626 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) | |
2 | f1ocnv 6626 | . . 3 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡◡𝐹:𝐴–1-1-onto→𝐵) | |
3 | dfrel2 6045 | . . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
4 | f1oeq1 6603 | . . . 4 ⊢ (◡◡𝐹 = 𝐹 → (◡◡𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵)) | |
5 | 3, 4 | sylbi 219 | . . 3 ⊢ (Rel 𝐹 → (◡◡𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵)) |
6 | 2, 5 | syl5ib 246 | . 2 ⊢ (Rel 𝐹 → (◡𝐹:𝐵–1-1-onto→𝐴 → 𝐹:𝐴–1-1-onto→𝐵)) |
7 | 1, 6 | impbid2 228 | 1 ⊢ (Rel 𝐹 → (𝐹:𝐴–1-1-onto→𝐵 ↔ ◡𝐹:𝐵–1-1-onto→𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ◡ccnv 5553 Rel wrel 5559 –1-1-onto→wf1o 6353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 |
This theorem is referenced by: hasheqf1oi 13711 |
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