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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 6568 | . 2 ⊢ (◡◡𝐴:dom 𝐴–1-1→V ↔ (◡◡𝐴:dom 𝐴⟶V ∧ Fun ◡◡◡𝐴)) | |
2 | dffn2 6739 | . . . 4 ⊢ (◡◡𝐴 Fn dom 𝐴 ↔ ◡◡𝐴:dom 𝐴⟶V) | |
3 | dmcnvcnv 5947 | . . . . 5 ⊢ dom ◡◡𝐴 = dom 𝐴 | |
4 | df-fn 6566 | . . . . 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 6125 | . . . . 5 ⊢ Rel ◡𝐴 | |
8 | dfrel2 6211 | . . . . 5 ⊢ (Rel ◡𝐴 ↔ ◡◡◡𝐴 = ◡𝐴) | |
9 | 7, 8 | mpbi 230 | . . . 4 ⊢ ◡◡◡𝐴 = ◡𝐴 |
10 | 9 | funeqi 6589 | . . 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 1537 Vcvv 3478 ◡ccnv 5688 dom cdm 5689 Rel wrel 5694 Fun wfun 6557 Fn wfn 6558 ⟶wf 6559 –1-1→wf1 6560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 |
This theorem is referenced by: (None) |
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