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Mirrors > Home > MPE Home > Th. List > f1cnvcnv | Unicode version |
Description: Two ways to express that a set (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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f1 5598 | . 2 | |
2 | dffn2 5737 | . . . 4 | |
3 | dmcnvcnv 5230 | . . . . 5 | |
4 | df-fn 5596 | . . . . 5 | |
5 | 3, 4 | mpbiran2 919 | . . . 4 |
6 | 2, 5 | bitr3i 251 | . . 3 |
7 | relcnv 5379 | . . . . 5 | |
8 | dfrel2 5462 | . . . . 5 | |
9 | 7, 8 | mpbi 208 | . . . 4 |
10 | 9 | funeqi 5613 | . . 3 |
11 | 6, 10 | anbi12ci 698 | . 2 |
12 | 1, 11 | bitri 249 | 1 |
Colors of variables: wff setvar class |
Syntax hints: <-> wb 184 /\ wa 369
= wceq 1395 cvv 3109
`' ccnv 5003 dom cdm 5004 Rel wrel 5009
Fun wfun 5587
Fn wfn 5588 --> wf 5589 -1-1-> wf1 5590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-sep 4573 ax-nul 4581 ax-pr 4691 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-ral 2812 df-rex 2813 df-rab 2816 df-v 3111 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-nul 3785 df-if 3942 df-sn 4030 df-pr 4032 df-op 4036 df-br 4453 df-opab 4511 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 |
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