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Theorem feq1 5367
Description: Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
feq1 (𝐹 = 𝐺 → (𝐹:𝐴𝐵𝐺:𝐴𝐵))

Proof of Theorem feq1
StepHypRef Expression
1 fneq1 5323 . . 3 (𝐹 = 𝐺 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))
2 rneq 4872 . . . 4 (𝐹 = 𝐺 → ran 𝐹 = ran 𝐺)
32sseq1d 3199 . . 3 (𝐹 = 𝐺 → (ran 𝐹𝐵 ↔ ran 𝐺𝐵))
41, 3anbi12d 473 . 2 (𝐹 = 𝐺 → ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺𝐵)))
5 df-f 5239 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
6 df-f 5239 . 2 (𝐺:𝐴𝐵 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺𝐵))
74, 5, 63bitr4g 223 1 (𝐹 = 𝐺 → (𝐹:𝐴𝐵𝐺:𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wss 3144  ran crn 4645   Fn wfn 5230  wf 5231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-fun 5237  df-fn 5238  df-f 5239
This theorem is referenced by:  feq1d  5371  feq1i  5377  f00  5426  f0bi  5427  f0dom0  5428  fconstg  5431  f1eq1  5435  fconst2g  5752  tfrcllemsucfn  6379  tfrcllemsucaccv  6380  tfrcllembxssdm  6382  tfrcllembfn  6383  tfrcllemex  6386  tfrcllemaccex  6387  tfrcllemres  6388  tfrcl  6390  elmapg  6688  ac6sfi  6927  updjud  7112  finomni  7169  exmidomni  7171  mkvprop  7187  1fv  10171  isgrpinv  13013  isghm  13199  upxp  14249  txcn  14252  dceqnconst  15287  dcapnconst  15288
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