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Mirrors > Home > ILE Home > Th. List > f1eq2 | GIF version |
Description: Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.) |
Ref | Expression |
---|---|
f1eq2 | ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐹:𝐵–1-1→𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq2 5368 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶)) | |
2 | 1 | anbi1d 465 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐹:𝐴⟶𝐶 ∧ Fun ◡𝐹) ↔ (𝐹:𝐵⟶𝐶 ∧ Fun ◡𝐹))) |
3 | df-f1 5240 | . 2 ⊢ (𝐹:𝐴–1-1→𝐶 ↔ (𝐹:𝐴⟶𝐶 ∧ Fun ◡𝐹)) | |
4 | df-f1 5240 | . 2 ⊢ (𝐹:𝐵–1-1→𝐶 ↔ (𝐹:𝐵⟶𝐶 ∧ Fun ◡𝐹)) | |
5 | 2, 3, 4 | 3bitr4g 223 | 1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐹:𝐵–1-1→𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ◡ccnv 4643 Fun wfun 5229 ⟶wf 5231 –1-1→wf1 5232 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-gen 1460 ax-4 1521 ax-17 1537 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-cleq 2182 df-fn 5238 df-f 5239 df-f1 5240 |
This theorem is referenced by: f1oeq2 5469 f1eq123d 5472 brdomg 6775 ennnfonelemen 12475 |
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