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Mirrors > Home > MPE Home > Th. List > f1oeq23 | Structured version Visualization version GIF version |
Description: Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.) |
Ref | Expression |
---|---|
f1oeq23 | ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1oeq2 6650 | . 2 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶)) | |
2 | f1oeq3 6651 | . 2 ⊢ (𝐶 = 𝐷 → (𝐹:𝐵–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐷)) | |
3 | 1, 2 | sylan9bb 513 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 –1-1-onto→wf1o 6379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3410 df-in 3873 df-ss 3883 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 |
This theorem is referenced by: f1ofvswap 7116 enfixsn 8754 ackbij2lem2 9854 seqf1o 13617 eulerthlem2 16335 isgim 18666 islmim 20099 fpwrelmapffs 30789 hgt750lemg 32346 poimirlem3 35517 poimirlem15 35529 eldioph2lem1 40285 fundcmpsurbijinj 44535 isomushgr 44951 |
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