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Mirrors > Home > MPE Home > Th. List > funcnv | Structured version Visualization version GIF version |
Description: The converse of a class is a function iff the class is single-rooted, which means that for any 𝑦 in the range of 𝐴 there is at most one 𝑥 such that 𝑥𝐴𝑦. Definition of single-rooted in [Enderton] p. 43. See funcnv2 5871 for a simpler version. (Contributed by NM, 13-Aug-2004.) |
Ref | Expression |
---|---|
funcnv | ⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3176 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | vex 3176 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | brelrn 5277 | . . . . . 6 ⊢ (𝑥𝐴𝑦 → 𝑦 ∈ ran 𝐴) |
4 | 3 | pm4.71ri 663 | . . . . 5 ⊢ (𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦)) |
5 | 4 | mobii 2481 | . . . 4 ⊢ (∃*𝑥 𝑥𝐴𝑦 ↔ ∃*𝑥(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦)) |
6 | moanimv 2519 | . . . 4 ⊢ (∃*𝑥(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦) ↔ (𝑦 ∈ ran 𝐴 → ∃*𝑥 𝑥𝐴𝑦)) | |
7 | 5, 6 | bitri 263 | . . 3 ⊢ (∃*𝑥 𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴 → ∃*𝑥 𝑥𝐴𝑦)) |
8 | 7 | albii 1737 | . 2 ⊢ (∀𝑦∃*𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(𝑦 ∈ ran 𝐴 → ∃*𝑥 𝑥𝐴𝑦)) |
9 | funcnv2 5871 | . 2 ⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦) | |
10 | df-ral 2901 | . 2 ⊢ (∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(𝑦 ∈ ran 𝐴 → ∃*𝑥 𝑥𝐴𝑦)) | |
11 | 8, 9, 10 | 3bitr4i 291 | 1 ⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 ∈ wcel 1977 ∃*wmo 2459 ∀wral 2896 class class class wbr 4583 ◡ccnv 5037 ran crn 5039 Fun wfun 5798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 |
This theorem is referenced by: funcnv3 5873 fncnv 5876 |
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