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| Mirrors > Home > MPE Home > Th. List > f1ocnvfv | Structured version Visualization version GIF version | ||
| Description: Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by Raph Levien, 10-Apr-2004.) |
| Ref | Expression |
|---|---|
| f1ocnvfv | ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝐹‘𝐶) = 𝐷 → (◡𝐹‘𝐷) = 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fveq2 6906 | . . 3 ⊢ (𝐷 = (𝐹‘𝐶) → (◡𝐹‘𝐷) = (◡𝐹‘(𝐹‘𝐶))) | |
| 2 | 1 | eqcoms 2745 | . 2 ⊢ ((𝐹‘𝐶) = 𝐷 → (◡𝐹‘𝐷) = (◡𝐹‘(𝐹‘𝐶))) |
| 3 | f1ocnvfv1 7296 | . . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝐶)) = 𝐶) | |
| 4 | 3 | eqeq2d 2748 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → ((◡𝐹‘𝐷) = (◡𝐹‘(𝐹‘𝐶)) ↔ (◡𝐹‘𝐷) = 𝐶)) |
| 5 | 2, 4 | imbitrid 244 | 1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝐹‘𝐶) = 𝐷 → (◡𝐹‘𝐷) = 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ◡ccnv 5684 –1-1-onto→wf1o 6560 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 |
| This theorem is referenced by: f1ocnvfvb 7299 f1oiso2 7372 curry1 8129 curry2 8132 dif1en 9200 dif1enOLD 9202 mapfienlem2 9446 infxpenc2lem1 10059 axcclem 10497 uzrdgfni 13999 uzrdgsuci 14001 fzennn 14009 axdc4uzlem 14024 seqf1olem1 14082 seqf1olem2 14083 hashginv 14373 sadaddlem 16503 xpsaddlem 17618 xpsvsca 17622 xpsle 17624 catcisolem 18155 mgmhmf1o 18713 mhmf1o 18809 ghmf1o 19266 lmhmf1o 21045 symgtgp 24114 xpsdsval 24391 noseqrdgfn 28312 noseqrdgsuc 28314 cnvbraval 32129 madjusmdetlem2 33827 reprpmtf1o 34641 derangenlem 35176 subfacp1lem4 35188 subfacp1lem5 35189 cvmliftlem9 35298 rngoisocnv 37988 cdleme51finvfvN 40557 ltrniotacnvval 40584 dssmapclsntr 44142 isubgr3stgrlem7 47939 |
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