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Mirrors > Home > ILE Home > Th. List > f1orescnv | GIF version |
Description: The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.) |
Ref | Expression |
---|---|
f1orescnv | ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → (◡𝐹 ↾ 𝑃):𝑃–1-1-onto→𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ocnv 5159 | . . 3 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → ◡(𝐹 ↾ 𝑅):𝑃–1-1-onto→𝑅) | |
2 | 1 | adantl 271 | . 2 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → ◡(𝐹 ↾ 𝑅):𝑃–1-1-onto→𝑅) |
3 | funcnvres 4992 | . . . 4 ⊢ (Fun ◡𝐹 → ◡(𝐹 ↾ 𝑅) = (◡𝐹 ↾ (𝐹 “ 𝑅))) | |
4 | df-ima 4376 | . . . . . 6 ⊢ (𝐹 “ 𝑅) = ran (𝐹 ↾ 𝑅) | |
5 | dff1o5 5155 | . . . . . . 7 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 ↔ ((𝐹 ↾ 𝑅):𝑅–1-1→𝑃 ∧ ran (𝐹 ↾ 𝑅) = 𝑃)) | |
6 | 5 | simprbi 269 | . . . . . 6 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → ran (𝐹 ↾ 𝑅) = 𝑃) |
7 | 4, 6 | syl5eq 2125 | . . . . 5 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → (𝐹 “ 𝑅) = 𝑃) |
8 | 7 | reseq2d 4630 | . . . 4 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → (◡𝐹 ↾ (𝐹 “ 𝑅)) = (◡𝐹 ↾ 𝑃)) |
9 | 3, 8 | sylan9eq 2133 | . . 3 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → ◡(𝐹 ↾ 𝑅) = (◡𝐹 ↾ 𝑃)) |
10 | f1oeq1 5137 | . . 3 ⊢ (◡(𝐹 ↾ 𝑅) = (◡𝐹 ↾ 𝑃) → (◡(𝐹 ↾ 𝑅):𝑃–1-1-onto→𝑅 ↔ (◡𝐹 ↾ 𝑃):𝑃–1-1-onto→𝑅)) | |
11 | 9, 10 | syl 14 | . 2 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → (◡(𝐹 ↾ 𝑅):𝑃–1-1-onto→𝑅 ↔ (◡𝐹 ↾ 𝑃):𝑃–1-1-onto→𝑅)) |
12 | 2, 11 | mpbid 145 | 1 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → (◡𝐹 ↾ 𝑃):𝑃–1-1-onto→𝑅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 ◡ccnv 4362 ran crn 4364 ↾ cres 4365 “ cima 4366 Fun wfun 4916 –1-1→wf1 4919 –1-1-onto→wf1o 4921 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 |
This theorem is referenced by: f1oresrab 5350 |
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