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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 5241. (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 5021 | . . 3 ⊢ Rel ◡𝐴 | |
2 | dffun2 5241 | . . 3 ⊢ (Fun ◡𝐴 ↔ (Rel ◡𝐴 ∧ ∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧))) | |
3 | 1, 2 | mpbiran 942 | . 2 ⊢ (Fun ◡𝐴 ↔ ∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧)) |
4 | alcom 1489 | . 2 ⊢ (∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧)) | |
5 | vex 2755 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
6 | vex 2755 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
7 | 5, 6 | brcnv 4825 | . . . . . 6 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
8 | vex 2755 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
9 | 5, 8 | brcnv 4825 | . . . . . 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 1482 | . . 3 ⊢ (∀𝑦∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧)) |
13 | 12 | albii 1481 | . 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 1362 class class class wbr 4018 ◡ccnv 4640 Rel wrel 4646 Fun wfun 5225 |
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 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-fun 5233 |
This theorem is referenced by: imain 5313 |
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