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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 5972 | . 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 206 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ↦ cmpt 5225 ran crn 5686 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-cnv 5693 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: fliftel 7329 oarec 8600 unfilem1 9343 elrest 17472 psgneldm2 19522 psgnfitr 19535 iscyggen2 19899 iscyg3 19904 cycsubgcyg 19919 eldprd 20024 leordtval2 23220 iocpnfordt 23223 icomnfordt 23224 lecldbas 23227 tsmsxplem1 24161 minveclem2 25460 lhop2 26054 taylthlem2 26416 taylthlem2OLD 26417 fsumvma 27257 dchrptlem2 27309 2sqlem1 27461 dchrisum0fno1 27555 minvecolem2 30894 swrdrn3 32940 nsgqusf1olem1 33441 nsgqusf1olem3 33443 rspectopn 33866 zarclsun 33869 zarcls 33873 gsumesum 34060 esumlub 34061 esumcst 34064 esumpcvgval 34079 esumgect 34091 esum2d 34094 sigapildsys 34163 sxbrsigalem2 34288 omssubaddlem 34301 omssubadd 34302 eulerpartgbij 34374 actfunsnf1o 34619 actfunsnrndisj 34620 reprsuc 34630 breprexplema 34645 bnj1366 34843 msubco 35536 msubvrs 35565 fin2so 37614 poimirlem17 37644 poimirlem20 37647 cntotbnd 37803 islsat 38992 |
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