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Mirrors > Home > ILE Home > Th. List > funcnv3 | GIF version |
Description: A condition showing a class is single-rooted. (See funcnv 5307). (Contributed by NM, 26-May-2006.) |
Ref | Expression |
---|---|
funcnv3 | ⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrn2 4844 | . . . . . 6 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} | |
2 | 1 | abeq2i 2304 | . . . . 5 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥 𝑥𝐴𝑦) |
3 | 2 | biimpi 120 | . . . 4 ⊢ (𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦) |
4 | 3 | biantrurd 305 | . . 3 ⊢ (𝑦 ∈ ran 𝐴 → (∃*𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))) |
5 | 4 | ralbiia 2508 | . 2 ⊢ (∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦)) |
6 | funcnv 5307 | . 2 ⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦) | |
7 | df-reu 2479 | . . . 4 ⊢ (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴 ∧ 𝑥𝐴𝑦)) | |
8 | vex 2763 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
9 | vex 2763 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | breldm 4860 | . . . . . 6 ⊢ (𝑥𝐴𝑦 → 𝑥 ∈ dom 𝐴) |
11 | 10 | pm4.71ri 392 | . . . . 5 ⊢ (𝑥𝐴𝑦 ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑥𝐴𝑦)) |
12 | 11 | eubii 2051 | . . . 4 ⊢ (∃!𝑥 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴 ∧ 𝑥𝐴𝑦)) |
13 | eu5 2089 | . . . 4 ⊢ (∃!𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦)) | |
14 | 7, 12, 13 | 3bitr2i 208 | . . 3 ⊢ (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦)) |
15 | 14 | ralbii 2500 | . 2 ⊢ (∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦)) |
16 | 5, 6, 15 | 3bitr4i 212 | 1 ⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∃wex 1503 ∃!weu 2042 ∃*wmo 2043 ∈ wcel 2164 ∀wral 2472 ∃!wreu 2474 class class class wbr 4029 ◡ccnv 4654 dom cdm 4655 ran crn 4656 Fun wfun 5240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-reu 2479 df-v 2762 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-br 4030 df-opab 4091 df-id 4322 df-xp 4661 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-fun 5248 |
This theorem is referenced by: (None) |
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