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Mirrors > Home > NFE Home > Th. List > eldm | GIF version |
Description: Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by set.mm contributors, 2-Apr-2004.) |
Ref | Expression |
---|---|
eldm | ⊢ (A ∈ dom B ↔ ∃y ABy) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dm 4788 | . . . 4 ⊢ dom B = ran ◡B | |
2 | 1 | eleq2i 2417 | . . 3 ⊢ (A ∈ dom B ↔ A ∈ ran ◡B) |
3 | elrn 4897 | . . 3 ⊢ (A ∈ ran ◡B ↔ ∃y y◡BA) | |
4 | 2, 3 | bitri 240 | . 2 ⊢ (A ∈ dom B ↔ ∃y y◡BA) |
5 | brcnv 4893 | . . 3 ⊢ (y◡BA ↔ ABy) | |
6 | 5 | exbii 1582 | . 2 ⊢ (∃y y◡BA ↔ ∃y ABy) |
7 | 4, 6 | bitri 240 | 1 ⊢ (A ∈ dom B ↔ ∃y ABy) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∃wex 1541 ∈ wcel 1710 class class class wbr 4640 ◡ccnv 4772 dom cdm 4773 ran crn 4774 |
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 4079 ax-xp 4080 ax-cnv 4081 ax-1c 4082 ax-sset 4083 ax-si 4084 ax-ins2 4085 ax-ins3 4086 ax-typlower 4087 ax-sn 4088 |
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 2479 df-ne 2519 df-ral 2620 df-rex 2621 df-reu 2622 df-rmo 2623 df-rab 2624 df-v 2862 df-sbc 3048 df-nin 3212 df-compl 3213 df-in 3214 df-un 3215 df-dif 3216 df-symdif 3217 df-ss 3260 df-pss 3262 df-nul 3552 df-if 3664 df-pw 3725 df-sn 3742 df-pr 3743 df-uni 3893 df-int 3928 df-opk 4059 df-1c 4137 df-pw1 4138 df-uni1 4139 df-xpk 4186 df-cnvk 4187 df-ins2k 4188 df-ins3k 4189 df-imak 4190 df-cok 4191 df-p6 4192 df-sik 4193 df-ssetk 4194 df-imagek 4195 df-idk 4196 df-iota 4340 df-0c 4378 df-addc 4379 df-nnc 4380 df-fin 4381 df-lefin 4441 df-ltfin 4442 df-ncfin 4443 df-tfin 4444 df-evenfin 4445 df-oddfin 4446 df-sfin 4447 df-spfin 4448 df-phi 4566 df-op 4567 df-proj1 4568 df-proj2 4569 df-opab 4624 df-br 4641 df-ima 4728 df-cnv 4786 df-rn 4787 df-dm 4788 |
This theorem is referenced by: eldm2 4900 dfdm2 4901 dfrn4 4905 dmun 4913 dmi 4920 dmcoss 4972 dmcosseq 4974 dmres 4987 dminss 5042 dmsnopg 5067 funeu 5132 dffun7 5134 dffun8 5135 fneu 5188 fnres 5200 ndmfv 5350 dmfco 5382 dff3 5421 dmsi 5520 dmtxp 5803 txpcofun 5804 dmpprod 5841 fvfullfunlem1 5862 erref 5960 erdmrn 5966 ecdmn0 5968 enpw1 6063 enprmaplem3 6079 dmfrec 6317 |
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