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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 5225. (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 5005 | . . 3 ⊢ Rel ◡𝐴 | |
2 | dffun2 5225 | . . 3 ⊢ (Fun ◡𝐴 ↔ (Rel ◡𝐴 ∧ ∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧))) | |
3 | 1, 2 | mpbiran 940 | . 2 ⊢ (Fun ◡𝐴 ↔ ∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧)) |
4 | alcom 1478 | . 2 ⊢ (∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧)) | |
5 | vex 2740 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
6 | vex 2740 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
7 | 5, 6 | brcnv 4809 | . . . . . 6 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
8 | vex 2740 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
9 | 5, 8 | brcnv 4809 | . . . . . 6 ⊢ (𝑦◡𝐴𝑧 ↔ 𝑧𝐴𝑦) |
10 | 7, 9 | anbi12i 460 | . . . . 5 ⊢ ((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) ↔ (𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦)) |
11 | 10 | imbi1i 238 | . . . 4 ⊢ (((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧)) |
12 | 11 | 2albii 1471 | . . 3 ⊢ (∀𝑦∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧)) |
13 | 12 | albii 1470 | . 2 ⊢ (∀𝑥∀𝑦∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧)) |
14 | 3, 4, 13 | 3bitri 206 | 1 ⊢ (Fun ◡𝐴 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∀wal 1351 class class class wbr 4002 ◡ccnv 4624 Rel wrel 4630 Fun wfun 5209 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4003 df-opab 4064 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-fun 5217 |
This theorem is referenced by: imain 5297 |
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