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Mirrors > Home > MPE Home > Th. List > elrnmpti | Structured version Visualization version GIF version |
Description: Membership in the range of a function. (Contributed by NM, 30-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
rnmpt.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
elrnmpti.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
elrnmpti | ⊢ (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrnmpti.2 | . . 3 ⊢ 𝐵 ∈ V | |
2 | 1 | rgenw 3065 | . 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ V |
3 | rnmpt.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | elrnmptg 5958 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) |
5 | 2, 4 | ax-mp 5 | 1 ⊢ (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ↦ cmpt 5231 ran crn 5677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-mpt 5232 df-cnv 5684 df-dm 5686 df-rn 5687 |
This theorem is referenced by: fliftel 7305 oarec 8561 unfilem1 9309 pwfilemOLD 9345 elrest 17372 psgneldm2 19371 psgnfitr 19384 iscyggen2 19748 iscyg3 19753 cycsubgcyg 19768 eldprd 19873 leordtval2 22715 iocpnfordt 22718 icomnfordt 22719 lecldbas 22722 tsmsxplem1 23656 minveclem2 24942 lhop2 25531 taylthlem2 25885 fsumvma 26713 dchrptlem2 26765 2sqlem1 26917 dchrisum0fno1 27011 minvecolem2 30123 swrdrn3 32114 nsgqusf1olem1 32519 nsgqusf1olem3 32521 rspectopn 32842 zarclsun 32845 zarcls 32849 gsumesum 33052 esumlub 33053 esumcst 33056 esumpcvgval 33071 esumgect 33083 esum2d 33086 sigapildsys 33155 sxbrsigalem2 33280 omssubaddlem 33293 omssubadd 33294 eulerpartgbij 33366 actfunsnf1o 33611 actfunsnrndisj 33612 reprsuc 33622 breprexplema 33637 bnj1366 33835 msubco 34517 msubvrs 34546 fin2so 36470 poimirlem17 36500 poimirlem20 36503 cntotbnd 36659 islsat 37856 |
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