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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 4850 | . 2 ⊢ (◡◡A:dom A–1-1→V ↔ (◡◡A:dom A⟶V ∧ Fun ◡◡◡A)) | |
2 | dffn2 4990 | . . . 4 ⊢ (◡◡A Fn dom A ↔ ◡◡A:dom A⟶V) | |
3 | dmcnvcnv 4501 | . . . . 5 ⊢ dom ◡◡A = dom A | |
4 | df-fn 4848 | . . . . 5 ⊢ (◡◡A Fn dom A ↔ (Fun ◡◡A ∧ dom ◡◡A = dom A)) | |
5 | 3, 4 | mpbiran2 847 | . . . 4 ⊢ (◡◡A Fn dom A ↔ Fun ◡◡A) |
6 | 2, 5 | bitr3i 175 | . . 3 ⊢ (◡◡A:dom A⟶V ↔ Fun ◡◡A) |
7 | relcnv 4646 | . . . . 5 ⊢ Rel ◡A | |
8 | dfrel2 4714 | . . . . 5 ⊢ (Rel ◡A ↔ ◡◡◡A = ◡A) | |
9 | 7, 8 | mpbi 133 | . . . 4 ⊢ ◡◡◡A = ◡A |
10 | 9 | funeqi 4865 | . . 3 ⊢ (Fun ◡◡◡A ↔ Fun ◡A) |
11 | 6, 10 | anbi12ci 434 | . 2 ⊢ ((◡◡A:dom A⟶V ∧ Fun ◡◡◡A) ↔ (Fun ◡A ∧ Fun ◡◡A)) |
12 | 1, 11 | bitri 173 | 1 ⊢ (◡◡A:dom A–1-1→V ↔ (Fun ◡A ∧ Fun ◡◡A)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 = wceq 1242 Vcvv 2551 ◡ccnv 4287 dom cdm 4288 Rel wrel 4293 Fun wfun 4839 Fn wfn 4840 ⟶wf 4841 –1-1→wf1 4842 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 |
This theorem is referenced by: (None) |
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