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Mirrors > Home > ILE Home > Th. List > feq3 | GIF version |
Description: Equality theorem for functions. (Contributed by NM, 1-Aug-1994.) |
Ref | Expression |
---|---|
feq3 | ⊢ (𝐴 = 𝐵 → (𝐹:𝐶⟶𝐴 ↔ 𝐹:𝐶⟶𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq2 3116 | . . 3 ⊢ (𝐴 = 𝐵 → (ran 𝐹 ⊆ 𝐴 ↔ ran 𝐹 ⊆ 𝐵)) | |
2 | 1 | anbi2d 459 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐹 Fn 𝐶 ∧ ran 𝐹 ⊆ 𝐴) ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 ⊆ 𝐵))) |
3 | df-f 5122 | . 2 ⊢ (𝐹:𝐶⟶𝐴 ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 ⊆ 𝐴)) | |
4 | df-f 5122 | . 2 ⊢ (𝐹:𝐶⟶𝐵 ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 ⊆ 𝐵)) | |
5 | 2, 3, 4 | 3bitr4g 222 | 1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶⟶𝐴 ↔ 𝐹:𝐶⟶𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ⊆ wss 3066 ran crn 4535 Fn wfn 5113 ⟶wf 5114 |
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 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-in 3072 df-ss 3079 df-f 5122 |
This theorem is referenced by: feq23 5253 feq3d 5256 feq123d 5258 fun2 5291 fconstg 5314 f1eq3 5320 fsng 5586 fsn2 5587 fsnunf 5613 mapvalg 6545 mapsn 6577 lmff 12407 txcn 12433 |
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