|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > funcnv2 | Structured version Visualization version GIF version | ||
| Description: A simpler equivalence for single-rooted (see funcnv 6635). (Contributed by NM, 9-Aug-2004.) | 
| Ref | Expression | 
|---|---|
| funcnv2 | ⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | relcnv 6122 | . . 3 ⊢ Rel ◡𝐴 | |
| 2 | dffun6 6574 | . . 3 ⊢ (Fun ◡𝐴 ↔ (Rel ◡𝐴 ∧ ∀𝑦∃*𝑥 𝑦◡𝐴𝑥)) | |
| 3 | 1, 2 | mpbiran 709 | . 2 ⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑦◡𝐴𝑥) | 
| 4 | vex 3484 | . . . . 5 ⊢ 𝑦 ∈ V | |
| 5 | vex 3484 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 6 | 4, 5 | brcnv 5893 | . . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) | 
| 7 | 6 | mobii 2548 | . . 3 ⊢ (∃*𝑥 𝑦◡𝐴𝑥 ↔ ∃*𝑥 𝑥𝐴𝑦) | 
| 8 | 7 | albii 1819 | . 2 ⊢ (∀𝑦∃*𝑥 𝑦◡𝐴𝑥 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦) | 
| 9 | 3, 8 | bitri 275 | 1 ⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∀wal 1538 ∃*wmo 2538 class class class wbr 5143 ◡ccnv 5684 Rel wrel 5690 Fun wfun 6555 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-fun 6563 | 
| This theorem is referenced by: funcnv 6635 fun2cnv 6637 fun11 6640 dff12 6803 1stconst 8125 2ndconst 8126 | 
| Copyright terms: Public domain | W3C validator |