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Mirrors > Home > ILE Home > Th. List > funcnveq | GIF version |
Description: Another way of expressing that a class is single-rooted. Counterpart to dffun2 4912. (Contributed by Jim Kingdon, 24-Dec-2018.) |
Ref | Expression |
---|---|
funcnveq | ⊢ (Fun ◡𝐴 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 4703 | . . 3 ⊢ Rel ◡𝐴 | |
2 | dffun2 4912 | . . 3 ⊢ (Fun ◡𝐴 ↔ (Rel ◡𝐴 ∧ ∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧))) | |
3 | 1, 2 | mpbiran 847 | . 2 ⊢ (Fun ◡𝐴 ↔ ∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧)) |
4 | alcom 1367 | . 2 ⊢ (∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧)) | |
5 | vex 2560 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
6 | vex 2560 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
7 | 5, 6 | brcnv 4518 | . . . . . 6 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
8 | vex 2560 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
9 | 5, 8 | brcnv 4518 | . . . . . 6 ⊢ (𝑦◡𝐴𝑧 ↔ 𝑧𝐴𝑦) |
10 | 7, 9 | anbi12i 433 | . . . . 5 ⊢ ((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) ↔ (𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦)) |
11 | 10 | imbi1i 227 | . . . 4 ⊢ (((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧)) |
12 | 11 | 2albii 1360 | . . 3 ⊢ (∀𝑦∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧)) |
13 | 12 | albii 1359 | . 2 ⊢ (∀𝑥∀𝑦∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧)) |
14 | 3, 4, 13 | 3bitri 195 | 1 ⊢ (Fun ◡𝐴 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1241 class class class wbr 3764 ◡ccnv 4344 Rel wrel 4350 Fun wfun 4896 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-fun 4904 |
This theorem is referenced by: imain 4981 |
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