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Mirrors > Home > ILE Home > Th. List > funcnv2 | GIF version |
Description: A simpler equivalence for single-rooted (see funcnv 5179). (Contributed by NM, 9-Aug-2004.) |
Ref | Expression |
---|---|
funcnv2 | ⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 4912 | . . 3 ⊢ Rel ◡𝐴 | |
2 | dffun6 5132 | . . 3 ⊢ (Fun ◡𝐴 ↔ (Rel ◡𝐴 ∧ ∀𝑦∃*𝑥 𝑦◡𝐴𝑥)) | |
3 | 1, 2 | mpbiran 924 | . 2 ⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑦◡𝐴𝑥) |
4 | vex 2684 | . . . . 5 ⊢ 𝑦 ∈ V | |
5 | vex 2684 | . . . . 5 ⊢ 𝑥 ∈ V | |
6 | 4, 5 | brcnv 4717 | . . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
7 | 6 | mobii 2034 | . . 3 ⊢ (∃*𝑥 𝑦◡𝐴𝑥 ↔ ∃*𝑥 𝑥𝐴𝑦) |
8 | 7 | albii 1446 | . 2 ⊢ (∀𝑦∃*𝑥 𝑦◡𝐴𝑥 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦) |
9 | 3, 8 | bitri 183 | 1 ⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∀wal 1329 ∃*wmo 1998 class class class wbr 3924 ◡ccnv 4533 Rel wrel 4539 Fun wfun 5112 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-fun 5120 |
This theorem is referenced by: funcnv 5179 fun2cnv 5182 fun11 5185 dff12 5322 1stconst 6111 2ndconst 6112 |
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