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Theorem feq2 5226
Description: Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
feq2 (𝐴 = 𝐵 → (𝐹:𝐴𝐶𝐹:𝐵𝐶))

Proof of Theorem feq2
StepHypRef Expression
1 fneq2 5182 . . 3 (𝐴 = 𝐵 → (𝐹 Fn 𝐴𝐹 Fn 𝐵))
21anbi1d 460 . 2 (𝐴 = 𝐵 → ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐶) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹𝐶)))
3 df-f 5097 . 2 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶))
4 df-f 5097 . 2 (𝐹:𝐵𝐶 ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹𝐶))
52, 3, 43bitr4g 222 1 (𝐴 = 𝐵 → (𝐹:𝐴𝐶𝐹:𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wss 3041  ran crn 4510   Fn wfn 5088  wf 5089
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-5 1408  ax-gen 1410  ax-4 1472  ax-17 1491  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-cleq 2110  df-fn 5096  df-f 5097
This theorem is referenced by:  feq23  5228  feq2d  5230  feq2i  5236  f00  5284  f0dom0  5286  f1eq2  5294  fressnfv  5575  tfrcllemsucfn  6218  tfrcllemsucaccv  6219  tfrcllembxssdm  6221  tfrcllembfn  6222  tfrcllemaccex  6226  tfrcllemres  6227  tfrcldm  6228  tfrcl  6229  mapvalg  6520  map0g  6550  ac6sfi  6760  isomni  6976  ismkv  6995
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