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Mirrors > Home > NFE Home > Th. List > f1eq2 | GIF version |
Description: Equality theorem for one-to-one functions. (Contributed by set.mm contributors, 10-Feb-1997.) |
Ref | Expression |
---|---|
f1eq2 | ⊢ (A = B → (F:A–1-1→C ↔ F:B–1-1→C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq2 5211 | . . 3 ⊢ (A = B → (F:A–→C ↔ F:B–→C)) | |
2 | 1 | anbi1d 685 | . 2 ⊢ (A = B → ((F:A–→C ∧ Fun ◡F) ↔ (F:B–→C ∧ Fun ◡F))) |
3 | df-f1 4792 | . 2 ⊢ (F:A–1-1→C ↔ (F:A–→C ∧ Fun ◡F)) | |
4 | df-f1 4792 | . 2 ⊢ (F:B–1-1→C ↔ (F:B–→C ∧ Fun ◡F)) | |
5 | 2, 3, 4 | 3bitr4g 279 | 1 ⊢ (A = B → (F:A–1-1→C ↔ F:B–1-1→C)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 = wceq 1642 ◡ccnv 4771 Fun wfun 4775 –→wf 4777 –1-1→wf1 4778 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-11 1746 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ex 1542 df-cleq 2346 df-fn 4790 df-f 4791 df-f1 4792 |
This theorem is referenced by: f1oeq2 5282 dflec3 6221 nclenc 6222 |
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