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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 3150 | . 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ V |
3 | rnmpt.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | elrnmptg 5831 | . 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 208 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ↦ cmpt 5146 ran crn 5556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-mpt 5147 df-cnv 5563 df-dm 5565 df-rn 5566 |
This theorem is referenced by: fliftel 7062 oarec 8188 unfilem1 8782 pwfilem 8818 elrest 16701 psgneldm2 18632 psgnfitr 18645 iscyggen2 19000 iscyg3 19005 cycsubgcyg 19021 eldprd 19126 leordtval2 21820 iocpnfordt 21823 icomnfordt 21824 lecldbas 21827 tsmsxplem1 22761 minveclem2 24029 lhop2 24612 taylthlem2 24962 fsumvma 25789 dchrptlem2 25841 2sqlem1 25993 dchrisum0fno1 26087 minvecolem2 28652 swrdrn3 30629 gsumesum 31318 esumlub 31319 esumcst 31322 esumpcvgval 31337 esumgect 31349 esum2d 31352 sigapildsys 31421 sxbrsigalem2 31544 omssubaddlem 31557 omssubadd 31558 eulerpartgbij 31630 actfunsnf1o 31875 actfunsnrndisj 31876 reprsuc 31886 breprexplema 31901 bnj1366 32101 msubco 32778 msubvrs 32807 fin2so 34894 poimirlem17 34924 poimirlem20 34927 cntotbnd 35089 islsat 36142 |
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