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Mirrors > Home > MPE Home > Th. List > f1oeq2 | Structured version Visualization version GIF version |
Description: Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.) |
Ref | Expression |
---|---|
f1oeq2 | ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1eq2 6783 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐹:𝐵–1-1→𝐶)) | |
2 | foeq2 6802 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–onto→𝐶 ↔ 𝐹:𝐵–onto→𝐶)) | |
3 | 1, 2 | anbi12d 631 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐹:𝐴–1-1→𝐶 ∧ 𝐹:𝐴–onto→𝐶) ↔ (𝐹:𝐵–1-1→𝐶 ∧ 𝐹:𝐵–onto→𝐶))) |
4 | df-f1o 6550 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐶 ↔ (𝐹:𝐴–1-1→𝐶 ∧ 𝐹:𝐴–onto→𝐶)) | |
5 | df-f1o 6550 | . 2 ⊢ (𝐹:𝐵–1-1-onto→𝐶 ↔ (𝐹:𝐵–1-1→𝐶 ∧ 𝐹:𝐵–onto→𝐶)) | |
6 | 3, 4, 5 | 3bitr4g 313 | 1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 –1-1→wf1 6540 –onto→wfo 6541 –1-1-onto→wf1o 6542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-cleq 2724 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 |
This theorem is referenced by: f1oeq23 6824 f1oeq123d 6827 f1oeq2d 6829 resin 6855 isoeq4 7316 breng 8947 brenOLD 8949 f1dmvrnfibi 9335 cnfcom 9694 infxpenc2 10016 fsumf1o 15668 sumsnf 15688 fprodf1o 15889 prodsn 15905 prodsnf 15907 znhash 21113 znunithash 21119 imasf1oxms 23997 wlksnwwlknvbij 29159 clwwlkvbij 29363 eupthp1 29466 derangval 34153 subfacp1lem2a 34166 subfacp1lem3 34168 subfacp1lem5 34170 sumsnd 43700 uspgrsprfo 46516 |
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