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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 5038. (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 4823 | . . 3 ⊢ Rel ◡𝐴 | |
2 | dffun2 5038 | . . 3 ⊢ (Fun ◡𝐴 ↔ (Rel ◡𝐴 ∧ ∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧))) | |
3 | 1, 2 | mpbiran 887 | . 2 ⊢ (Fun ◡𝐴 ↔ ∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧)) |
4 | alcom 1413 | . 2 ⊢ (∀𝑦∀𝑥∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧)) | |
5 | vex 2623 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
6 | vex 2623 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
7 | 5, 6 | brcnv 4632 | . . . . . 6 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦) |
8 | vex 2623 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
9 | 5, 8 | brcnv 4632 | . . . . . 6 ⊢ (𝑦◡𝐴𝑧 ↔ 𝑧𝐴𝑦) |
10 | 7, 9 | anbi12i 449 | . . . . 5 ⊢ ((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) ↔ (𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦)) |
11 | 10 | imbi1i 237 | . . . 4 ⊢ (((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧)) |
12 | 11 | 2albii 1406 | . . 3 ⊢ (∀𝑦∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧)) |
13 | 12 | albii 1405 | . 2 ⊢ (∀𝑥∀𝑦∀𝑧((𝑦◡𝐴𝑥 ∧ 𝑦◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧)) |
14 | 3, 4, 13 | 3bitri 205 | 1 ⊢ (Fun ◡𝐴 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1288 class class class wbr 3851 ◡ccnv 4451 Rel wrel 4457 Fun wfun 5022 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-br 3852 df-opab 3906 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-fun 5030 |
This theorem is referenced by: imain 5109 |
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