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Theorem feq3 5032
 Description: Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
feq3 (𝐴 = 𝐵 → (𝐹:𝐶𝐴𝐹:𝐶𝐵))

Proof of Theorem feq3
StepHypRef Expression
1 sseq2 2967 . . 3 (𝐴 = 𝐵 → (ran 𝐹𝐴 ↔ ran 𝐹𝐵))
21anbi2d 437 . 2 (𝐴 = 𝐵 → ((𝐹 Fn 𝐶 ∧ ran 𝐹𝐴) ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹𝐵)))
3 df-f 4906 . 2 (𝐹:𝐶𝐴 ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹𝐴))
4 df-f 4906 . 2 (𝐹:𝐶𝐵 ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹𝐵))
52, 3, 43bitr4g 212 1 (𝐴 = 𝐵 → (𝐹:𝐶𝐴𝐹:𝐶𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ⊆ wss 2917  ran crn 4346   Fn wfn 4897  ⟶wf 4898 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-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931  df-f 4906 This theorem is referenced by:  feq23  5033  feq123d  5037  fun2  5064  fconstg  5083  f1eq3  5089  fsng  5336  fsn2  5337  fsnunf  5362
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