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Mirrors > Home > MPE Home > Th. List > dffun5 | Structured version Visualization version GIF version |
Description: Alternate definition of function. (Contributed by NM, 29-Dec-1996.) |
Ref | Expression |
---|---|
dffun5 | ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffun3 6548 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))) | |
2 | df-br 5140 | . . . . . . 7 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴) | |
3 | 2 | imbi1i 349 | . . . . . 6 ⊢ ((𝑥𝐴𝑦 → 𝑦 = 𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧)) |
4 | 3 | albii 1813 | . . . . 5 ⊢ (∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧) ↔ ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧)) |
5 | 4 | exbii 1842 | . . . 4 ⊢ (∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧) ↔ ∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧)) |
6 | 5 | albii 1813 | . . 3 ⊢ (∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧) ↔ ∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧)) |
7 | 6 | anbi2i 622 | . 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)) ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧))) |
8 | 1, 7 | bitri 275 | 1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1531 ∃wex 1773 ∈ wcel 2098 ⟨cop 4627 class class class wbr 5139 Rel wrel 5672 Fun wfun 6528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-mo 2526 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-fun 6536 |
This theorem is referenced by: funimaexgOLD 6626 fvn0ssdmfun 7067 uzrdgfni 13924 noseqrdgfn 28120 dffrege115 43279 |
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