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Mirrors > Home > ILE Home > Th. List > fun2cnv | GIF version |
Description: The double converse of a class is a function iff the class is single-valued. Each side is equivalent to Definition 6.4(2) of [TakeutiZaring] p. 23, who use the notation "Un(A)" for single-valued. Note that 𝐴 is not necessarily a function. (Contributed by NM, 13-Aug-2004.) |
Ref | Expression |
---|---|
fun2cnv | ⊢ (Fun ◡◡𝐴 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcnv2 5191 | . 2 ⊢ (Fun ◡◡𝐴 ↔ ∀𝑥∃*𝑦 𝑦◡𝐴𝑥) | |
2 | vex 2692 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | vex 2692 | . . . . 5 ⊢ 𝑥 ∈ V | |
4 | 2, 3 | brcnv 4730 | . . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
5 | 4 | mobii 2037 | . . 3 ⊢ (∃*𝑦 𝑦◡𝐴𝑥 ↔ ∃*𝑦 𝑥𝐴𝑦) |
6 | 5 | albii 1447 | . 2 ⊢ (∀𝑥∃*𝑦 𝑦◡𝐴𝑥 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦) |
7 | 1, 6 | bitri 183 | 1 ⊢ (Fun ◡◡𝐴 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∀wal 1330 ∃*wmo 2001 class class class wbr 3937 ◡ccnv 4546 Fun wfun 5125 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-fun 5133 |
This theorem is referenced by: svrelfun 5196 fun11 5198 |
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