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Mirrors > Home > ILE Home > Th. List > funcnv | GIF version |
Description: The converse of a class is a function iff the class is single-rooted, which means that for any y in the range of A there is at most one x such that xAy. Definition of single-rooted in [Enderton] p. 43. See funcnv2 4902 for a simpler version. (Contributed by NM, 13-Aug-2004.) |
Ref | Expression |
---|---|
funcnv | ⊢ (Fun ◡A ↔ ∀y ∈ ran A∃*x xAy) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2554 | . . . . . . 7 ⊢ x ∈ V | |
2 | vex 2554 | . . . . . . 7 ⊢ y ∈ V | |
3 | 1, 2 | brelrn 4510 | . . . . . 6 ⊢ (xAy → y ∈ ran A) |
4 | 3 | pm4.71ri 372 | . . . . 5 ⊢ (xAy ↔ (y ∈ ran A ∧ xAy)) |
5 | 4 | mobii 1934 | . . . 4 ⊢ (∃*x xAy ↔ ∃*x(y ∈ ran A ∧ xAy)) |
6 | moanimv 1972 | . . . 4 ⊢ (∃*x(y ∈ ran A ∧ xAy) ↔ (y ∈ ran A → ∃*x xAy)) | |
7 | 5, 6 | bitri 173 | . . 3 ⊢ (∃*x xAy ↔ (y ∈ ran A → ∃*x xAy)) |
8 | 7 | albii 1356 | . 2 ⊢ (∀y∃*x xAy ↔ ∀y(y ∈ ran A → ∃*x xAy)) |
9 | funcnv2 4902 | . 2 ⊢ (Fun ◡A ↔ ∀y∃*x xAy) | |
10 | df-ral 2305 | . 2 ⊢ (∀y ∈ ran A∃*x xAy ↔ ∀y(y ∈ ran A → ∃*x xAy)) | |
11 | 8, 9, 10 | 3bitr4i 201 | 1 ⊢ (Fun ◡A ↔ ∀y ∈ ran A∃*x xAy) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1240 ∈ wcel 1390 ∃*wmo 1898 ∀wral 2300 class class class wbr 3755 ◡ccnv 4287 ran crn 4289 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-rex 2306 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-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-fun 4847 |
This theorem is referenced by: funcnv3 4904 fncnv 4908 |
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