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Theorem f1ocnvfv 5362
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.)
Assertion
Ref Expression
f1ocnvfv ((𝐹:A1-1-ontoB 𝐶 A) → ((𝐹𝐶) = 𝐷 → (𝐹𝐷) = 𝐶))

Proof of Theorem f1ocnvfv
StepHypRef Expression
1 fveq2 5121 . . 3 (𝐷 = (𝐹𝐶) → (𝐹𝐷) = (𝐹‘(𝐹𝐶)))
21eqcoms 2040 . 2 ((𝐹𝐶) = 𝐷 → (𝐹𝐷) = (𝐹‘(𝐹𝐶)))
3 f1ocnvfv1 5360 . . 3 ((𝐹:A1-1-ontoB 𝐶 A) → (𝐹‘(𝐹𝐶)) = 𝐶)
43eqeq2d 2048 . 2 ((𝐹:A1-1-ontoB 𝐶 A) → ((𝐹𝐷) = (𝐹‘(𝐹𝐶)) ↔ (𝐹𝐷) = 𝐶))
52, 4syl5ib 143 1 ((𝐹:A1-1-ontoB 𝐶 A) → ((𝐹𝐶) = 𝐷 → (𝐹𝐷) = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  ccnv 4287  1-1-ontowf1o 4844  cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853
This theorem is referenced by:  f1ocnvfvb  5363  f1oiso2  5409  frecuzrdgfn  8879  frecuzrdgsuc  8882  frecfzennn  8884
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