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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 2742 | . 2 ⊢ ((𝐹‘𝐶) = 𝐷 → (◡𝐹‘𝐷) = (◡𝐹‘(𝐹‘𝐶))) |
3 | f1ocnvfv1 7295 | . . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝐶)) = 𝐶) | |
4 | 3 | eqeq2d 2745 | . 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 1536 ∈ wcel 2105 ◡ccnv 5687 –1-1-onto→wf1o 6561 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 |
This theorem is referenced by: f1ocnvfvb 7298 f1oiso2 7371 curry1 8127 curry2 8130 dif1en 9198 dif1enOLD 9200 mapfienlem2 9443 infxpenc2lem1 10056 axcclem 10494 uzrdgfni 13995 uzrdgsuci 13997 fzennn 14005 axdc4uzlem 14020 seqf1olem1 14078 seqf1olem2 14079 hashginv 14369 sadaddlem 16499 xpsaddlem 17619 xpsvsca 17623 xpsle 17625 catcisolem 18163 mgmhmf1o 18725 mhmf1o 18821 ghmf1o 19278 lmhmf1o 21062 symgtgp 24129 xpsdsval 24406 noseqrdgfn 28326 noseqrdgsuc 28328 cnvbraval 32138 madjusmdetlem2 33788 reprpmtf1o 34619 derangenlem 35155 subfacp1lem4 35167 subfacp1lem5 35168 cvmliftlem9 35277 rngoisocnv 37967 cdleme51finvfvN 40537 ltrniotacnvval 40564 dssmapclsntr 44118 isubgr3stgrlem7 47874 |
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