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| Mirrors > Home > MPE Home > Th. List > f1cnvcnv | Structured version Visualization version GIF version | ||
| Description: Two ways to express that a set 𝐴 (not necessarily a function) is one-to-one. Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one. We do not introduce a separate notation since we rarely use it. (Contributed by NM, 13-Aug-2004.) |
| Ref | Expression |
|---|---|
| f1cnvcnv | ⊢ (◡◡𝐴:dom 𝐴–1-1→V ↔ (Fun ◡𝐴 ∧ Fun ◡◡𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-f1 6566 | . 2 ⊢ (◡◡𝐴:dom 𝐴–1-1→V ↔ (◡◡𝐴:dom 𝐴⟶V ∧ Fun ◡◡◡𝐴)) | |
| 2 | dffn2 6738 | . . . 4 ⊢ (◡◡𝐴 Fn dom 𝐴 ↔ ◡◡𝐴:dom 𝐴⟶V) | |
| 3 | dmcnvcnv 5944 | . . . . 5 ⊢ dom ◡◡𝐴 = dom 𝐴 | |
| 4 | df-fn 6564 | . . . . 5 ⊢ (◡◡𝐴 Fn dom 𝐴 ↔ (Fun ◡◡𝐴 ∧ dom ◡◡𝐴 = dom 𝐴)) | |
| 5 | 3, 4 | mpbiran2 710 | . . . 4 ⊢ (◡◡𝐴 Fn dom 𝐴 ↔ Fun ◡◡𝐴) |
| 6 | 2, 5 | bitr3i 277 | . . 3 ⊢ (◡◡𝐴:dom 𝐴⟶V ↔ Fun ◡◡𝐴) |
| 7 | relcnv 6122 | . . . . 5 ⊢ Rel ◡𝐴 | |
| 8 | dfrel2 6209 | . . . . 5 ⊢ (Rel ◡𝐴 ↔ ◡◡◡𝐴 = ◡𝐴) | |
| 9 | 7, 8 | mpbi 230 | . . . 4 ⊢ ◡◡◡𝐴 = ◡𝐴 |
| 10 | 9 | funeqi 6587 | . . 3 ⊢ (Fun ◡◡◡𝐴 ↔ Fun ◡𝐴) |
| 11 | 6, 10 | anbi12ci 629 | . 2 ⊢ ((◡◡𝐴:dom 𝐴⟶V ∧ Fun ◡◡◡𝐴) ↔ (Fun ◡𝐴 ∧ Fun ◡◡𝐴)) |
| 12 | 1, 11 | bitri 275 | 1 ⊢ (◡◡𝐴:dom 𝐴–1-1→V ↔ (Fun ◡𝐴 ∧ Fun ◡◡𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 Vcvv 3480 ◡ccnv 5684 dom cdm 5685 Rel wrel 5690 Fun wfun 6555 Fn wfn 6556 ⟶wf 6557 –1-1→wf1 6558 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 |
| This theorem is referenced by: (None) |
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