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Theorem f1ocnvfvb 6532
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 6531 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴) → ((𝐹𝐶) = 𝐷 → (𝐹𝐷) = 𝐶))
213adant3 1080 . 2 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((𝐹𝐶) = 𝐷 → (𝐹𝐷) = 𝐶))
3 fveq2 6189 . . . . 5 (𝐶 = (𝐹𝐷) → (𝐹𝐶) = (𝐹‘(𝐹𝐷)))
43eqcoms 2629 . . . 4 ((𝐹𝐷) = 𝐶 → (𝐹𝐶) = (𝐹‘(𝐹𝐷)))
5 f1ocnvfv2 6530 . . . . 5 ((𝐹:𝐴1-1-onto𝐵𝐷𝐵) → (𝐹‘(𝐹𝐷)) = 𝐷)
65eqeq2d 2631 . . . 4 ((𝐹:𝐴1-1-onto𝐵𝐷𝐵) → ((𝐹𝐶) = (𝐹‘(𝐹𝐷)) ↔ (𝐹𝐶) = 𝐷))
74, 6syl5ib 234 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐷𝐵) → ((𝐹𝐷) = 𝐶 → (𝐹𝐶) = 𝐷))
873adant2 1079 . 2 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((𝐹𝐷) = 𝐶 → (𝐹𝐶) = 𝐷))
92, 8impbid 202 1 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((𝐹𝐶) = 𝐷 ↔ (𝐹𝐷) = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1037   = wceq 1482  wcel 1989  ccnv 5111  1-1-ontowf1o 5885  cfv 5886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894
This theorem is referenced by:  f1ofveu  6642  f1ocnvfv3  6643  1arith2  15626  f1omvdcnv  17858  f1omvdconj  17860  txhmeo  21600  iccpnfcnv  22737  dvcnvlem  23733  logeftb  24324  sqff1o  24902  bracnlnval  28957  cdlemg17h  35782
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