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Theorem feq1 5259
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 5215 . . 3 (𝐹 = 𝐺 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))
2 rneq 4770 . . . 4 (𝐹 = 𝐺 → ran 𝐹 = ran 𝐺)
32sseq1d 3127 . . 3 (𝐹 = 𝐺 → (ran 𝐹𝐵 ↔ ran 𝐺𝐵))
41, 3anbi12d 465 . 2 (𝐹 = 𝐺 → ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺𝐵)))
5 df-f 5131 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
6 df-f 5131 . 2 (𝐺:𝐴𝐵 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺𝐵))
74, 5, 63bitr4g 222 1 (𝐹 = 𝐺 → (𝐹:𝐴𝐵𝐺:𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wss 3072  ran crn 4544   Fn wfn 5122  wf 5123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2689  df-un 3076  df-in 3078  df-ss 3085  df-sn 3534  df-pr 3535  df-op 3537  df-br 3934  df-opab 3994  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-fun 5129  df-fn 5130  df-f 5131
This theorem is referenced by:  feq1d  5263  feq1i  5269  f00  5318  f0bi  5319  f0dom0  5320  fconstg  5323  f1eq1  5327  fconst2g  5639  tfrcllemsucfn  6254  tfrcllemsucaccv  6255  tfrcllembxssdm  6257  tfrcllembfn  6258  tfrcllemex  6261  tfrcllemaccex  6262  tfrcllemres  6263  tfrcl  6265  elmapg  6559  ac6sfi  6796  updjud  6971  finomni  7016  exmidomni  7018  mkvprop  7036  1fv  9943  upxp  12471  txcn  12474  dcapncf  13406
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