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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 6891 | . . 3 ⊢ (𝐷 = (𝐹‘𝐶) → (◡𝐹‘𝐷) = (◡𝐹‘(𝐹‘𝐶))) | |
2 | 1 | eqcoms 2740 | . 2 ⊢ ((𝐹‘𝐶) = 𝐷 → (◡𝐹‘𝐷) = (◡𝐹‘(𝐹‘𝐶))) |
3 | f1ocnvfv1 7273 | . . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝐶)) = 𝐶) | |
4 | 3 | eqeq2d 2743 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → ((◡𝐹‘𝐷) = (◡𝐹‘(𝐹‘𝐶)) ↔ (◡𝐹‘𝐷) = 𝐶)) |
5 | 2, 4 | imbitrid 243 | 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 396 = wceq 1541 ∈ wcel 2106 ◡ccnv 5675 –1-1-onto→wf1o 6542 ‘cfv 6543 |
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-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 |
This theorem is referenced by: f1ocnvfvb 7276 f1oiso2 7348 curry1 8089 curry2 8092 dif1en 9159 dif1enOLD 9161 mapfienlem2 9400 infxpenc2lem1 10013 axcclem 10451 uzrdgfni 13922 uzrdgsuci 13924 fzennn 13932 axdc4uzlem 13947 seqf1olem1 14006 seqf1olem2 14007 hashginv 14293 sadaddlem 16406 xpsaddlem 17518 xpsvsca 17522 xpsle 17524 catcisolem 18059 mhmf1o 18681 ghmf1o 19121 lmhmf1o 20656 symgtgp 23609 xpsdsval 23886 cnvbraval 31358 madjusmdetlem2 32803 reprpmtf1o 33633 derangenlem 34157 subfacp1lem4 34169 subfacp1lem5 34170 cvmliftlem9 34279 rngoisocnv 36844 cdleme51finvfvN 39421 ltrniotacnvval 39448 dssmapclsntr 42870 mgmhmf1o 46547 |
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