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Mirrors > Home > MPE Home > Th. List > fun2cnv | Structured version Visualization version 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 6494 | . 2 ⊢ (Fun ◡◡𝐴 ↔ ∀𝑥∃*𝑦 𝑦◡𝐴𝑥) | |
2 | vex 3433 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | vex 3433 | . . . . 5 ⊢ 𝑥 ∈ V | |
4 | 2, 3 | brcnv 5784 | . . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
5 | 4 | mobii 2548 | . . 3 ⊢ (∃*𝑦 𝑦◡𝐴𝑥 ↔ ∃*𝑦 𝑥𝐴𝑦) |
6 | 5 | albii 1822 | . 2 ⊢ (∀𝑥∃*𝑦 𝑦◡𝐴𝑥 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦) |
7 | 1, 6 | bitri 274 | 1 ⊢ (Fun ◡◡𝐴 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wal 1537 ∃*wmo 2538 class class class wbr 5073 ◡ccnv 5583 Fun wfun 6420 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pr 5350 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rab 3073 df-v 3431 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5074 df-opab 5136 df-id 5484 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-fun 6428 |
This theorem is referenced by: svrelfun 6498 fun11 6500 |
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