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Mirrors > Home > ILE Home > Th. List > f1cnvcnv | 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 5123 | . 2 ⊢ (◡◡𝐴:dom 𝐴–1-1→V ↔ (◡◡𝐴:dom 𝐴⟶V ∧ Fun ◡◡◡𝐴)) | |
2 | dffn2 5269 | . . . 4 ⊢ (◡◡𝐴 Fn dom 𝐴 ↔ ◡◡𝐴:dom 𝐴⟶V) | |
3 | dmcnvcnv 4758 | . . . . 5 ⊢ dom ◡◡𝐴 = dom 𝐴 | |
4 | df-fn 5121 | . . . . 5 ⊢ (◡◡𝐴 Fn dom 𝐴 ↔ (Fun ◡◡𝐴 ∧ dom ◡◡𝐴 = dom 𝐴)) | |
5 | 3, 4 | mpbiran2 925 | . . . 4 ⊢ (◡◡𝐴 Fn dom 𝐴 ↔ Fun ◡◡𝐴) |
6 | 2, 5 | bitr3i 185 | . . 3 ⊢ (◡◡𝐴:dom 𝐴⟶V ↔ Fun ◡◡𝐴) |
7 | relcnv 4912 | . . . . 5 ⊢ Rel ◡𝐴 | |
8 | dfrel2 4984 | . . . . 5 ⊢ (Rel ◡𝐴 ↔ ◡◡◡𝐴 = ◡𝐴) | |
9 | 7, 8 | mpbi 144 | . . . 4 ⊢ ◡◡◡𝐴 = ◡𝐴 |
10 | 9 | funeqi 5139 | . . 3 ⊢ (Fun ◡◡◡𝐴 ↔ Fun ◡𝐴) |
11 | 6, 10 | anbi12ci 456 | . 2 ⊢ ((◡◡𝐴:dom 𝐴⟶V ∧ Fun ◡◡◡𝐴) ↔ (Fun ◡𝐴 ∧ Fun ◡◡𝐴)) |
12 | 1, 11 | bitri 183 | 1 ⊢ (◡◡𝐴:dom 𝐴–1-1→V ↔ (Fun ◡𝐴 ∧ Fun ◡◡𝐴)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1331 Vcvv 2681 ◡ccnv 4533 dom cdm 4534 Rel wrel 4539 Fun wfun 5112 Fn wfn 5113 ⟶wf 5114 –1-1→wf1 5115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 |
This theorem is referenced by: (None) |
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