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Mirrors > Home > NFE Home > Th. List > dfrn4 | GIF version |
Description: Alternate definition of range. (Contributed by set.mm contributors, 5-Feb-2015.) |
Ref | Expression |
---|---|
dfrn4 | ⊢ ran A = dom ◡A |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brcnv 4892 | . . . 4 ⊢ (x◡Ay ↔ yAx) | |
2 | 1 | exbii 1582 | . . 3 ⊢ (∃y x◡Ay ↔ ∃y yAx) |
3 | eldm 4898 | . . 3 ⊢ (x ∈ dom ◡A ↔ ∃y x◡Ay) | |
4 | elrn 4896 | . . 3 ⊢ (x ∈ ran A ↔ ∃y yAx) | |
5 | 2, 3, 4 | 3bitr4ri 269 | . 2 ⊢ (x ∈ ran A ↔ x ∈ dom ◡A) |
6 | 5 | eqriv 2350 | 1 ⊢ ran A = dom ◡A |
Colors of variables: wff setvar class |
Syntax hints: ∃wex 1541 = wceq 1642 ∈ wcel 1710 class class class wbr 4639 ◡ccnv 4771 dom cdm 4772 ran crn 4773 |
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-ima 4727 df-cnv 4785 df-rn 4786 df-dm 4787 |
This theorem is referenced by: rnss 4959 brelrn 4960 rncoss 4972 rncoeq 4975 cnvimarndm 5017 rnun 5036 rnin 5037 rnxp 5051 rnxpss 5053 fncnv 5158 funcnvres 5165 funimacnv 5168 fimacnvdisj 5244 dff1o4 5294 foimacnv 5303 f1ococnv2 5309 rnsi 5521 rnpprod 5842 enpw1 6062 enmap2 6068 |
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