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Theorem f1ocnvfvb 7273
Description: Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.)
Assertion
Ref Expression
f1ocnvfvb ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((𝐹𝐶) = 𝐷 ↔ (𝐹𝐷) = 𝐶))

Proof of Theorem f1ocnvfvb
StepHypRef Expression
1 f1ocnvfv 7272 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴) → ((𝐹𝐶) = 𝐷 → (𝐹𝐷) = 𝐶))
213adant3 1129 . 2 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((𝐹𝐶) = 𝐷 → (𝐹𝐷) = 𝐶))
3 fveq2 6885 . . . . 5 (𝐶 = (𝐹𝐷) → (𝐹𝐶) = (𝐹‘(𝐹𝐷)))
43eqcoms 2734 . . . 4 ((𝐹𝐷) = 𝐶 → (𝐹𝐶) = (𝐹‘(𝐹𝐷)))
5 f1ocnvfv2 7271 . . . . 5 ((𝐹:𝐴1-1-onto𝐵𝐷𝐵) → (𝐹‘(𝐹𝐷)) = 𝐷)
65eqeq2d 2737 . . . 4 ((𝐹:𝐴1-1-onto𝐵𝐷𝐵) → ((𝐹𝐶) = (𝐹‘(𝐹𝐷)) ↔ (𝐹𝐶) = 𝐷))
74, 6imbitrid 243 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐷𝐵) → ((𝐹𝐷) = 𝐶 → (𝐹𝐶) = 𝐷))
873adant2 1128 . 2 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((𝐹𝐷) = 𝐶 → (𝐹𝐶) = 𝐷))
92, 8impbid 211 1 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((𝐹𝐶) = 𝐷 ↔ (𝐹𝐷) = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  ccnv 5668  1-1-ontowf1o 6536  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545
This theorem is referenced by:  f1ofveu  7399  f1ocnvfv3  7400  1arith2  16870  f1omvdcnv  19364  f1omvdconj  19366  rngqiprngu  21171  txhmeo  23662  iccpnfcnv  24824  dvcnvlem  25863  logeftb  26472  sqff1o  27069  bracnlnval  31876  cdlemg17h  40052  isomuspgrlem1  47067
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