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Theorem feq1 4973
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 4930 . . 3 (𝐹 = 𝐺 → (𝐹 Fn A𝐺 Fn A))
2 rneq 4504 . . . 4 (𝐹 = 𝐺 → ran 𝐹 = ran 𝐺)
32sseq1d 2966 . . 3 (𝐹 = 𝐺 → (ran 𝐹B ↔ ran 𝐺B))
41, 3anbi12d 442 . 2 (𝐹 = 𝐺 → ((𝐹 Fn A ran 𝐹B) ↔ (𝐺 Fn A ran 𝐺B)))
5 df-f 4849 . 2 (𝐹:AB ↔ (𝐹 Fn A ran 𝐹B))
6 df-f 4849 . 2 (𝐺:AB ↔ (𝐺 Fn A ran 𝐺B))
74, 5, 63bitr4g 212 1 (𝐹 = 𝐺 → (𝐹:AB𝐺:AB))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wss 2911  ran crn 4289   Fn wfn 4840  wf 4841
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
This theorem is referenced by:  feq1d  4977  feq1i  4982  f00  5024  fconstg  5026  f1eq1  5030  fconst2g  5319  1fv  8766
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