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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 3066 | . 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ V |
3 | rnmpt.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | elrnmptg 5956 | . 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 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ↦ 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 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 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 7301 oarec 8558 unfilem1 9306 pwfilemOLD 9342 elrest 17369 psgneldm2 19365 psgnfitr 19378 iscyggen2 19741 iscyg3 19746 cycsubgcyg 19761 eldprd 19866 leordtval2 22698 iocpnfordt 22701 icomnfordt 22702 lecldbas 22705 tsmsxplem1 23639 minveclem2 24925 lhop2 25514 taylthlem2 25868 fsumvma 26696 dchrptlem2 26748 2sqlem1 26900 dchrisum0fno1 26994 minvecolem2 30106 swrdrn3 32097 nsgqusf1olem1 32487 nsgqusf1olem3 32489 rspectopn 32785 zarclsun 32788 zarcls 32792 gsumesum 32995 esumlub 32996 esumcst 32999 esumpcvgval 33014 esumgect 33026 esum2d 33029 sigapildsys 33098 sxbrsigalem2 33223 omssubaddlem 33236 omssubadd 33237 eulerpartgbij 33309 actfunsnf1o 33554 actfunsnrndisj 33555 reprsuc 33565 breprexplema 33580 bnj1366 33778 msubco 34460 msubvrs 34489 fin2so 36413 poimirlem17 36443 poimirlem20 36446 cntotbnd 36602 islsat 37799 |
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