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Mirrors > Home > NFE 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 5155 for a simpler version. (Contributed by set.mm contributors, 13-Aug-2004.) |
Ref | Expression |
---|---|
funcnv | ⊢ (Fun ◡A ↔ ∀y ∈ ran A∃*x xAy) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brelrn 4960 | . . . . . 6 ⊢ (xAy → y ∈ ran A) | |
2 | 1 | pm4.71ri 614 | . . . . 5 ⊢ (xAy ↔ (y ∈ ran A ∧ xAy)) |
3 | 2 | mobii 2240 | . . . 4 ⊢ (∃*x xAy ↔ ∃*x(y ∈ ran A ∧ xAy)) |
4 | moanimv 2262 | . . . 4 ⊢ (∃*x(y ∈ ran A ∧ xAy) ↔ (y ∈ ran A → ∃*x xAy)) | |
5 | 3, 4 | bitri 240 | . . 3 ⊢ (∃*x xAy ↔ (y ∈ ran A → ∃*x xAy)) |
6 | 5 | albii 1566 | . 2 ⊢ (∀y∃*x xAy ↔ ∀y(y ∈ ran A → ∃*x xAy)) |
7 | funcnv2 5155 | . 2 ⊢ (Fun ◡A ↔ ∀y∃*x xAy) | |
8 | df-ral 2619 | . 2 ⊢ (∀y ∈ ran A∃*x xAy ↔ ∀y(y ∈ ran A → ∃*x xAy)) | |
9 | 6, 7, 8 | 3bitr4i 268 | 1 ⊢ (Fun ◡A ↔ ∀y ∈ ran A∃*x xAy) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∀wal 1540 ∈ wcel 1710 ∃*wmo 2205 ∀wral 2614 class class class wbr 4639 ◡ccnv 4771 ran crn 4773 Fun wfun 4775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-co 4726 df-ima 4727 df-id 4767 df-cnv 4785 df-rn 4786 df-dm 4787 df-fun 4789 |
This theorem is referenced by: funcnv3 5157 fncnv 5158 |
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