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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 6187 | . 2 ⊢ (◡◡𝐴:dom 𝐴–1-1→V ↔ (◡◡𝐴:dom 𝐴⟶V ∧ Fun ◡◡◡𝐴)) | |
2 | dffn2 6340 | . . . 4 ⊢ (◡◡𝐴 Fn dom 𝐴 ↔ ◡◡𝐴:dom 𝐴⟶V) | |
3 | dmcnvcnv 5640 | . . . . 5 ⊢ dom ◡◡𝐴 = dom 𝐴 | |
4 | df-fn 6185 | . . . . 5 ⊢ (◡◡𝐴 Fn dom 𝐴 ↔ (Fun ◡◡𝐴 ∧ dom ◡◡𝐴 = dom 𝐴)) | |
5 | 3, 4 | mpbiran2 697 | . . . 4 ⊢ (◡◡𝐴 Fn dom 𝐴 ↔ Fun ◡◡𝐴) |
6 | 2, 5 | bitr3i 269 | . . 3 ⊢ (◡◡𝐴:dom 𝐴⟶V ↔ Fun ◡◡𝐴) |
7 | relcnv 5801 | . . . . 5 ⊢ Rel ◡𝐴 | |
8 | dfrel2 5880 | . . . . 5 ⊢ (Rel ◡𝐴 ↔ ◡◡◡𝐴 = ◡𝐴) | |
9 | 7, 8 | mpbi 222 | . . . 4 ⊢ ◡◡◡𝐴 = ◡𝐴 |
10 | 9 | funeqi 6203 | . . 3 ⊢ (Fun ◡◡◡𝐴 ↔ Fun ◡𝐴) |
11 | 6, 10 | anbi12ci 618 | . 2 ⊢ ((◡◡𝐴:dom 𝐴⟶V ∧ Fun ◡◡◡𝐴) ↔ (Fun ◡𝐴 ∧ Fun ◡◡𝐴)) |
12 | 1, 11 | bitri 267 | 1 ⊢ (◡◡𝐴:dom 𝐴–1-1→V ↔ (Fun ◡𝐴 ∧ Fun ◡◡𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 = wceq 1507 Vcvv 3409 ◡ccnv 5400 dom cdm 5401 Rel wrel 5406 Fun wfun 6176 Fn wfn 6177 ⟶wf 6178 –1-1→wf1 6179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5054 ax-nul 5061 ax-pr 5180 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-rab 3091 df-v 3411 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-br 4924 df-opab 4986 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 |
This theorem is referenced by: (None) |
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