ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f1eq1 Structured version   GIF version

Theorem f1eq1 5030
Description: Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
Assertion
Ref Expression
f1eq1 (𝐹 = 𝐺 → (𝐹:A1-1B𝐺:A1-1B))

Proof of Theorem f1eq1
StepHypRef Expression
1 feq1 4973 . . 3 (𝐹 = 𝐺 → (𝐹:AB𝐺:AB))
2 cnveq 4452 . . . 4 (𝐹 = 𝐺𝐹 = 𝐺)
32funeqd 4866 . . 3 (𝐹 = 𝐺 → (Fun 𝐹 ↔ Fun 𝐺))
41, 3anbi12d 442 . 2 (𝐹 = 𝐺 → ((𝐹:AB Fun 𝐹) ↔ (𝐺:AB Fun 𝐺)))
5 df-f1 4850 . 2 (𝐹:A1-1B ↔ (𝐹:AB Fun 𝐹))
6 df-f1 4850 . 2 (𝐺:A1-1B ↔ (𝐺:AB Fun 𝐺))
74, 5, 63bitr4g 212 1 (𝐹 = 𝐺 → (𝐹:A1-1B𝐺:A1-1B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  ccnv 4287  Fun wfun 4839  wf 4841  1-1wf1 4842
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-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850
This theorem is referenced by:  f1oeq1  5060  f1eq123d  5064  fun11iun  5090  fo00  5105  tposf12  5825  f1dom2g  6172  f1domg  6174  dom3d  6190  domtr  6201
  Copyright terms: Public domain W3C validator