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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 6425). (Contributed by NM, 9-Aug-2004.) |
Ref | Expression |
---|---|
funcnv2 | ⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 5969 | . . 3 ⊢ Rel ◡𝐴 | |
2 | dffun6 6372 | . . 3 ⊢ (Fun ◡𝐴 ↔ (Rel ◡𝐴 ∧ ∀𝑦∃*𝑥 𝑦◡𝐴𝑥)) | |
3 | 1, 2 | mpbiran 707 | . 2 ⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑦◡𝐴𝑥) |
4 | vex 3499 | . . . . 5 ⊢ 𝑦 ∈ V | |
5 | vex 3499 | . . . . 5 ⊢ 𝑥 ∈ V | |
6 | 4, 5 | brcnv 5755 | . . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
7 | 6 | mobii 2631 | . . 3 ⊢ (∃*𝑥 𝑦◡𝐴𝑥 ↔ ∃*𝑥 𝑥𝐴𝑦) |
8 | 7 | albii 1820 | . 2 ⊢ (∀𝑦∃*𝑥 𝑦◡𝐴𝑥 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦) |
9 | 3, 8 | bitri 277 | 1 ⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1535 ∃*wmo 2620 class class class wbr 5068 ◡ccnv 5556 Rel wrel 5562 Fun wfun 6351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-fun 6359 |
This theorem is referenced by: funcnv 6425 fun2cnv 6427 fun11 6430 dff12 6576 1stconst 7797 2ndconst 7798 |
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