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Mirrors > Home > MPE Home > Th. List > dffun4 | Structured version Visualization version GIF version |
Description: Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.) |
Ref | Expression |
---|---|
dffun4 | ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ 𝐴) → 𝑦 = 𝑧))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffun2 5814 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))) | |
2 | df-br 4584 | . . . . . . 7 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
3 | df-br 4584 | . . . . . . 7 ⊢ (𝑥𝐴𝑧 ↔ 〈𝑥, 𝑧〉 ∈ 𝐴) | |
4 | 2, 3 | anbi12i 729 | . . . . . 6 ⊢ ((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ 𝐴)) |
5 | 4 | imbi1i 338 | . . . . 5 ⊢ (((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ 𝐴) → 𝑦 = 𝑧)) |
6 | 5 | albii 1737 | . . . 4 ⊢ (∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ 𝐴) → 𝑦 = 𝑧)) |
7 | 6 | 2albii 1738 | . . 3 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ 𝐴) → 𝑦 = 𝑧)) |
8 | 7 | anbi2i 726 | . 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)) ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ 𝐴) → 𝑦 = 𝑧))) |
9 | 1, 8 | bitri 263 | 1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ 𝐴) → 𝑦 = 𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 ∈ wcel 1977 〈cop 4131 class class class wbr 4583 Rel wrel 5043 Fun wfun 5798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-cnv 5046 df-co 5047 df-fun 5806 |
This theorem is referenced by: funopg 5836 funun 5846 fununi 5878 tfrlem7 7366 hashfun 13084 bnj1379 30155 dffun10 31191 elfuns 31192 |
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