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Mirrors > Home > MPE Home > Th. List > funcnv2 | Structured version Visualization version GIF version |
Description: A simpler equivalence for single-rooted (see funcnv 6568). (Contributed by NM, 9-Aug-2004.) |
Ref | Expression |
---|---|
funcnv2 | ⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 6055 | . . 3 ⊢ Rel ◡𝐴 | |
2 | dffun6 6507 | . . 3 ⊢ (Fun ◡𝐴 ↔ (Rel ◡𝐴 ∧ ∀𝑦∃*𝑥 𝑦◡𝐴𝑥)) | |
3 | 1, 2 | mpbiran 707 | . 2 ⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑦◡𝐴𝑥) |
4 | vex 3448 | . . . . 5 ⊢ 𝑦 ∈ V | |
5 | vex 3448 | . . . . 5 ⊢ 𝑥 ∈ V | |
6 | 4, 5 | brcnv 5837 | . . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
7 | 6 | mobii 2546 | . . 3 ⊢ (∃*𝑥 𝑦◡𝐴𝑥 ↔ ∃*𝑥 𝑥𝐴𝑦) |
8 | 7 | albii 1821 | . 2 ⊢ (∀𝑦∃*𝑥 𝑦◡𝐴𝑥 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦) |
9 | 3, 8 | bitri 274 | 1 ⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wal 1539 ∃*wmo 2536 class class class wbr 5104 ◡ccnv 5631 Rel wrel 5637 Fun wfun 6488 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-mo 2538 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-br 5105 df-opab 5167 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-fun 6496 |
This theorem is referenced by: funcnv 6568 fun2cnv 6570 fun11 6573 dff12 6735 1stconst 8029 2ndconst 8030 |
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