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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 6843 | . . 3 ⊢ (𝐷 = (𝐹‘𝐶) → (◡𝐹‘𝐷) = (◡𝐹‘(𝐹‘𝐶))) | |
2 | 1 | eqcoms 2741 | . 2 ⊢ ((𝐹‘𝐶) = 𝐷 → (◡𝐹‘𝐷) = (◡𝐹‘(𝐹‘𝐶))) |
3 | f1ocnvfv1 7223 | . . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝐶)) = 𝐶) | |
4 | 3 | eqeq2d 2744 | . 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 397 = wceq 1542 ∈ wcel 2107 ◡ccnv 5633 –1-1-onto→wf1o 6496 ‘cfv 6497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 |
This theorem is referenced by: f1ocnvfvb 7226 f1oiso2 7298 curry1 8037 curry2 8040 dif1en 9107 dif1enOLD 9109 mapfienlem2 9347 infxpenc2lem1 9960 axcclem 10398 uzrdgfni 13869 uzrdgsuci 13871 fzennn 13879 axdc4uzlem 13894 seqf1olem1 13953 seqf1olem2 13954 hashginv 14240 sadaddlem 16351 xpsaddlem 17460 xpsvsca 17464 xpsle 17466 catcisolem 18001 mhmf1o 18617 ghmf1o 19043 lmhmf1o 20522 symgtgp 23473 xpsdsval 23750 cnvbraval 31094 madjusmdetlem2 32466 reprpmtf1o 33296 derangenlem 33822 subfacp1lem4 33834 subfacp1lem5 33835 cvmliftlem9 33944 rngoisocnv 36486 cdleme51finvfvN 39064 ltrniotacnvval 39091 dssmapclsntr 42489 mgmhmf1o 46167 |
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