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Mirrors > Home > ILE Home > Th. List > dffun8 | GIF version |
Description: Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 4871. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
dffun8 | ⊢ (Fun A ↔ (Rel A ∧ ∀x ∈ dom A∃!y xAy)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffun7 4871 | . 2 ⊢ (Fun A ↔ (Rel A ∧ ∀x ∈ dom A∃*y xAy)) | |
2 | df-mo 1901 | . . . . 5 ⊢ (∃*y xAy ↔ (∃y xAy → ∃!y xAy)) | |
3 | vex 2554 | . . . . . . 7 ⊢ x ∈ V | |
4 | 3 | eldm 4475 | . . . . . 6 ⊢ (x ∈ dom A ↔ ∃y xAy) |
5 | pm5.5 231 | . . . . . 6 ⊢ (∃y xAy → ((∃y xAy → ∃!y xAy) ↔ ∃!y xAy)) | |
6 | 4, 5 | sylbi 114 | . . . . 5 ⊢ (x ∈ dom A → ((∃y xAy → ∃!y xAy) ↔ ∃!y xAy)) |
7 | 2, 6 | syl5bb 181 | . . . 4 ⊢ (x ∈ dom A → (∃*y xAy ↔ ∃!y xAy)) |
8 | 7 | ralbiia 2332 | . . 3 ⊢ (∀x ∈ dom A∃*y xAy ↔ ∀x ∈ dom A∃!y xAy) |
9 | 8 | anbi2i 430 | . 2 ⊢ ((Rel A ∧ ∀x ∈ dom A∃*y xAy) ↔ (Rel A ∧ ∀x ∈ dom A∃!y xAy)) |
10 | 1, 9 | bitri 173 | 1 ⊢ (Fun A ↔ (Rel A ∧ ∀x ∈ dom A∃!y xAy)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∃wex 1378 ∈ wcel 1390 ∃!weu 1897 ∃*wmo 1898 ∀wral 2300 class class class wbr 3755 dom cdm 4288 Rel wrel 4293 Fun wfun 4839 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-id 4021 df-cnv 4296 df-co 4297 df-dm 4298 df-fun 4847 |
This theorem is referenced by: funco 4883 funimaexglem 4925 funfveu 5131 |
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