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Mirrors > Home > MPE Home > Th. List > f1oeq3 | Structured version Visualization version GIF version |
Description: Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.) |
Ref | Expression |
---|---|
f1oeq3 | ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:𝐶–1-1-onto→𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1eq3 6011 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1→𝐴 ↔ 𝐹:𝐶–1-1→𝐵)) | |
2 | foeq3 6026 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–onto→𝐴 ↔ 𝐹:𝐶–onto→𝐵)) | |
3 | 1, 2 | anbi12d 743 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐹:𝐶–1-1→𝐴 ∧ 𝐹:𝐶–onto→𝐴) ↔ (𝐹:𝐶–1-1→𝐵 ∧ 𝐹:𝐶–onto→𝐵))) |
4 | df-f1o 5811 | . 2 ⊢ (𝐹:𝐶–1-1-onto→𝐴 ↔ (𝐹:𝐶–1-1→𝐴 ∧ 𝐹:𝐶–onto→𝐴)) | |
5 | df-f1o 5811 | . 2 ⊢ (𝐹:𝐶–1-1-onto→𝐵 ↔ (𝐹:𝐶–1-1→𝐵 ∧ 𝐹:𝐶–onto→𝐵)) | |
6 | 3, 4, 5 | 3bitr4g 302 | 1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:𝐶–1-1-onto→𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 –1-1→wf1 5801 –onto→wfo 5802 –1-1-onto→wf1o 5803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-in 3547 df-ss 3554 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 |
This theorem is referenced by: f1oeq23 6043 f1oeq123d 6046 f1oeq3d 6047 f1ores 6064 resin 6071 f1osng 6089 f1oresrab 6302 fveqf1o 6457 isoeq5 6471 isoini2 6489 ncanth 6509 oacomf1o 7532 mapsnf1o 7835 bren 7850 xpcomf1o 7934 domss2 8004 isinf 8058 wemapwe 8477 oef1o 8478 cnfcomlem 8479 cnfcom2 8482 cnfcom3 8484 cnfcom3clem 8485 infxpenc 8724 infxpenc2lem1 8725 infxpenc2 8728 fin1a2lem6 9110 hsmexlem1 9131 pwfseqlem5 9364 pwfseq 9365 hashgf1o 12632 axdc4uzlem 12644 sumeq1 14267 fsumss 14303 fsumcnv 14346 prodeq1f 14477 fprodss 14517 fprodcnv 14552 unbenlem 15450 4sqlem11 15497 pwssnf1o 15981 catcisolem 16579 equivestrcsetc 16615 yoniso 16748 gsumvalx 17093 gsumpropd 17095 gsumpropd2lem 17096 xpsmnd 17153 xpsgrp 17357 cayley 17657 cayleyth 17658 gsumval3lem1 18129 gsumval3lem2 18130 gsumcom2 18197 scmatrngiso 20161 m2cpmrngiso 20382 cncfcnvcn 22532 ovolicc2lem4 23095 logf1o2 24196 uslgraf1oedg 25888 clwwlkvbij 26329 adjbd1o 28328 rinvf1o 28814 eulerpartgbij 29761 eulerpartlemgh 29767 derangval 30403 subfacp1lem2a 30416 subfacp1lem3 30418 subfacp1lem5 30420 mrsubff1o 30666 msubff1o 30708 bj-finsumval0 32324 f1omptsnlem 32359 f1omptsn 32360 poimirlem4 32583 poimirlem9 32588 poimirlem15 32594 ismtyval 32769 ismrer1 32807 rngoisoval 32946 lautset 34386 pautsetN 34402 hvmap1o2 36072 pwfi2f1o 36684 imasgim 36688 uspgrf1oedg 40403 usgrf1oedg 40434 clwwlksvbij 41229 |
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