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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 5899 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))) | |
2 | df-br 4654 | . . . . . . 7 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
3 | 2 | imbi1i 339 | . . . . . 6 ⊢ ((𝑥𝐴𝑦 → 𝑦 = 𝑧) ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧)) |
4 | 3 | albii 1747 | . . . . 5 ⊢ (∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧) ↔ ∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧)) |
5 | 4 | exbii 1774 | . . . 4 ⊢ (∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧) ↔ ∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧)) |
6 | 5 | albii 1747 | . . 3 ⊢ (∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧) ↔ ∀𝑥∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧)) |
7 | 6 | anbi2i 730 | . 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)) ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧))) |
8 | 1, 7 | bitri 264 | 1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∀wal 1481 ∃wex 1704 ∈ wcel 1990 〈cop 4183 class class class wbr 4653 Rel wrel 5119 Fun wfun 5882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-cnv 5122 df-co 5123 df-fun 5890 |
This theorem is referenced by: funimaexg 5975 fvn0ssdmfun 6350 uzrdgfni 12757 dffrege115 38272 |
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