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Mirrors > Home > ILE Home > Th. List > f1eq3 | GIF version |
Description: Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.) |
Ref | Expression |
---|---|
f1eq3 | ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1→𝐴 ↔ 𝐹:𝐶–1-1→𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq3 5057 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶⟶𝐴 ↔ 𝐹:𝐶⟶𝐵)) | |
2 | 1 | anbi1d 453 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐹:𝐶⟶𝐴 ∧ Fun ◡𝐹) ↔ (𝐹:𝐶⟶𝐵 ∧ Fun ◡𝐹))) |
3 | df-f1 4931 | . 2 ⊢ (𝐹:𝐶–1-1→𝐴 ↔ (𝐹:𝐶⟶𝐴 ∧ Fun ◡𝐹)) | |
4 | df-f1 4931 | . 2 ⊢ (𝐹:𝐶–1-1→𝐵 ↔ (𝐹:𝐶⟶𝐵 ∧ Fun ◡𝐹)) | |
5 | 2, 3, 4 | 3bitr4g 221 | 1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1→𝐴 ↔ 𝐹:𝐶–1-1→𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1285 ◡ccnv 4364 Fun wfun 4920 ⟶wf 4922 –1-1→wf1 4923 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-11 1438 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 |
This theorem depends on definitions: df-bi 115 df-nf 1391 df-sb 1687 df-clab 2069 df-cleq 2075 df-clel 2078 df-in 2980 df-ss 2987 df-f 4930 df-f1 4931 |
This theorem is referenced by: f1oeq3 5144 f1eq123d 5146 tposf12 5912 brdomg 6288 |
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