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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 3063 | . 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ V |
3 | rnmpt.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | elrnmptg 5957 | . 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 1539 ∈ wcel 2104 ∀wral 3059 ∃wrex 3068 Vcvv 3472 ↦ cmpt 5230 ran crn 5676 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-mpt 5231 df-cnv 5683 df-dm 5685 df-rn 5686 |
This theorem is referenced by: fliftel 7308 oarec 8564 unfilem1 9312 pwfilemOLD 9348 elrest 17377 psgneldm2 19413 psgnfitr 19426 iscyggen2 19790 iscyg3 19795 cycsubgcyg 19810 eldprd 19915 leordtval2 22936 iocpnfordt 22939 icomnfordt 22940 lecldbas 22943 tsmsxplem1 23877 minveclem2 25174 lhop2 25767 taylthlem2 26122 fsumvma 26952 dchrptlem2 27004 2sqlem1 27156 dchrisum0fno1 27250 minvecolem2 30395 swrdrn3 32386 nsgqusf1olem1 32798 nsgqusf1olem3 32800 rspectopn 33145 zarclsun 33148 zarcls 33152 gsumesum 33355 esumlub 33356 esumcst 33359 esumpcvgval 33374 esumgect 33386 esum2d 33389 sigapildsys 33458 sxbrsigalem2 33583 omssubaddlem 33596 omssubadd 33597 eulerpartgbij 33669 actfunsnf1o 33914 actfunsnrndisj 33915 reprsuc 33925 breprexplema 33940 bnj1366 34138 msubco 34820 msubvrs 34849 fin2so 36778 poimirlem17 36808 poimirlem20 36811 cntotbnd 36967 islsat 38164 |
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