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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 6663 | . . 3 ⊢ (𝐷 = (𝐹‘𝐶) → (◡𝐹‘𝐷) = (◡𝐹‘(𝐹‘𝐶))) | |
2 | 1 | eqcoms 2827 | . 2 ⊢ ((𝐹‘𝐶) = 𝐷 → (◡𝐹‘𝐷) = (◡𝐹‘(𝐹‘𝐶))) |
3 | f1ocnvfv1 7025 | . . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝐶)) = 𝐶) | |
4 | 3 | eqeq2d 2830 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → ((◡𝐹‘𝐷) = (◡𝐹‘(𝐹‘𝐶)) ↔ (◡𝐹‘𝐷) = 𝐶)) |
5 | 2, 4 | syl5ib 246 | 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 398 = wceq 1531 ∈ wcel 2108 ◡ccnv 5547 –1-1-onto→wf1o 6347 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 |
This theorem is referenced by: f1ocnvfvb 7028 f1oiso2 7097 curry1 7791 curry2 7794 mapfienlem2 8861 infxpenc2lem1 9437 axcclem 9871 uzrdgfni 13318 uzrdgsuci 13320 fzennn 13328 axdc4uzlem 13343 seqf1olem1 13401 seqf1olem2 13402 hashginv 13686 sadaddlem 15807 xpsaddlem 16838 xpsvsca 16842 xpsle 16844 catcisolem 17358 mhmf1o 17958 ghmf1o 18380 lmhmf1o 19810 symgtgp 22706 xpsdsval 22983 cnvbraval 29879 madjusmdetlem2 31086 reprpmtf1o 31890 derangenlem 32411 subfacp1lem4 32423 subfacp1lem5 32424 cvmliftlem9 32533 rngoisocnv 35251 cdleme51finvfvN 37683 ltrniotacnvval 37710 dssmapclsntr 40470 mgmhmf1o 44045 |
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