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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 6637). (Contributed by NM, 9-Aug-2004.) |
Ref | Expression |
---|---|
funcnv2 | ⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 6125 | . . 3 ⊢ Rel ◡𝐴 | |
2 | dffun6 6576 | . . 3 ⊢ (Fun ◡𝐴 ↔ (Rel ◡𝐴 ∧ ∀𝑦∃*𝑥 𝑦◡𝐴𝑥)) | |
3 | 1, 2 | mpbiran 709 | . 2 ⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑦◡𝐴𝑥) |
4 | vex 3482 | . . . . 5 ⊢ 𝑦 ∈ V | |
5 | vex 3482 | . . . . 5 ⊢ 𝑥 ∈ V | |
6 | 4, 5 | brcnv 5896 | . . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
7 | 6 | mobii 2546 | . . 3 ⊢ (∃*𝑥 𝑦◡𝐴𝑥 ↔ ∃*𝑥 𝑥𝐴𝑦) |
8 | 7 | albii 1816 | . 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 1535 ∃*wmo 2536 class class class wbr 5148 ◡ccnv 5688 Rel wrel 5694 Fun wfun 6557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-fun 6565 |
This theorem is referenced by: funcnv 6637 fun2cnv 6639 fun11 6642 dff12 6804 1stconst 8124 2ndconst 8125 |
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