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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 6392 | . 2 ⊢ (Fun ◡◡𝐴 ↔ ∀𝑥∃*𝑦 𝑦◡𝐴𝑥) | |
2 | vex 3444 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | vex 3444 | . . . . 5 ⊢ 𝑥 ∈ V | |
4 | 2, 3 | brcnv 5717 | . . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
5 | 4 | mobii 2606 | . . 3 ⊢ (∃*𝑦 𝑦◡𝐴𝑥 ↔ ∃*𝑦 𝑥𝐴𝑦) |
6 | 5 | albii 1821 | . 2 ⊢ (∀𝑥∃*𝑦 𝑦◡𝐴𝑥 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦) |
7 | 1, 6 | bitri 278 | 1 ⊢ (Fun ◡◡𝐴 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∀wal 1536 ∃*wmo 2596 class class class wbr 5030 ◡ccnv 5518 Fun wfun 6318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-fun 6326 |
This theorem is referenced by: svrelfun 6396 fun11 6398 |
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