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Mirrors > Home > ILE Home > Th. List > dffun7 | GIF version |
Description: Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. (Enderton's definition is ambiguous because "there is only one" could mean either "there is at most one" or "there is exactly one". However, dffun8 5245 shows that it does not matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.) |
Ref | Expression |
---|---|
dffun7 | ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffun6 5231 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦)) | |
2 | moabs 2075 | . . . . . 6 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ (∃𝑦 𝑥𝐴𝑦 → ∃*𝑦 𝑥𝐴𝑦)) | |
3 | vex 2741 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
4 | 3 | eldm 4825 | . . . . . . 7 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑥𝐴𝑦) |
5 | 4 | imbi1i 238 | . . . . . 6 ⊢ ((𝑥 ∈ dom 𝐴 → ∃*𝑦 𝑥𝐴𝑦) ↔ (∃𝑦 𝑥𝐴𝑦 → ∃*𝑦 𝑥𝐴𝑦)) |
6 | 2, 5 | bitr4i 187 | . . . . 5 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ (𝑥 ∈ dom 𝐴 → ∃*𝑦 𝑥𝐴𝑦)) |
7 | 6 | albii 1470 | . . . 4 ⊢ (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥(𝑥 ∈ dom 𝐴 → ∃*𝑦 𝑥𝐴𝑦)) |
8 | df-ral 2460 | . . . 4 ⊢ (∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥(𝑥 ∈ dom 𝐴 → ∃*𝑦 𝑥𝐴𝑦)) | |
9 | 7, 8 | bitr4i 187 | . . 3 ⊢ (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦) |
10 | 9 | anbi2i 457 | . 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦)) |
11 | 1, 10 | bitri 184 | 1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∀wal 1351 ∃wex 1492 ∃*wmo 2027 ∈ wcel 2148 ∀wral 2455 class class class wbr 4004 dom cdm 4627 Rel wrel 4632 Fun wfun 5211 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2740 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-br 4005 df-opab 4066 df-id 4294 df-cnv 4635 df-co 4636 df-dm 4637 df-fun 5219 |
This theorem is referenced by: dffun8 5245 dffun9 5246 funco 5257 funimaexglem 5300 frecuzrdgtcl 10412 frecuzrdgfunlem 10419 imasaddfnlemg 12735 |
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