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Mirrors > Home > ILE Home > Th. List > funcnv3 | GIF version |
Description: A condition showing a class is single-rooted. (See funcnv 4980). (Contributed by NM, 26-May-2006.) |
Ref | Expression |
---|---|
funcnv3 | ⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrn2 4541 | . . . . . 6 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} | |
2 | 1 | abeq2i 2189 | . . . . 5 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥 𝑥𝐴𝑦) |
3 | 2 | biimpi 118 | . . . 4 ⊢ (𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦) |
4 | 3 | biantrurd 299 | . . 3 ⊢ (𝑦 ∈ ran 𝐴 → (∃*𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))) |
5 | 4 | ralbiia 2380 | . 2 ⊢ (∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦)) |
6 | funcnv 4980 | . 2 ⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦) | |
7 | df-reu 2355 | . . . 4 ⊢ (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴 ∧ 𝑥𝐴𝑦)) | |
8 | vex 2604 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
9 | vex 2604 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | breldm 4557 | . . . . . 6 ⊢ (𝑥𝐴𝑦 → 𝑥 ∈ dom 𝐴) |
11 | 10 | pm4.71ri 384 | . . . . 5 ⊢ (𝑥𝐴𝑦 ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑥𝐴𝑦)) |
12 | 11 | eubii 1950 | . . . 4 ⊢ (∃!𝑥 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴 ∧ 𝑥𝐴𝑦)) |
13 | eu5 1988 | . . . 4 ⊢ (∃!𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦)) | |
14 | 7, 12, 13 | 3bitr2i 206 | . . 3 ⊢ (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦)) |
15 | 14 | ralbii 2372 | . 2 ⊢ (∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦)) |
16 | 5, 6, 15 | 3bitr4i 210 | 1 ⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 ∃wex 1421 ∈ wcel 1433 ∃!weu 1941 ∃*wmo 1942 ∀wral 2348 ∃!wreu 2350 class class class wbr 3785 ◡ccnv 4362 dom cdm 4363 ran crn 4364 Fun wfun 4916 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-reu 2355 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-fun 4924 |
This theorem is referenced by: (None) |
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