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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 2908 | . 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ V |
3 | rnmpt.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | elrnmptg 5296 | . 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 195 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ↦ cmpt 4643 ran crn 5039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-mpt 4645 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: fliftel 6459 oarec 7529 unfilem1 8109 pwfilem 8143 elrest 15911 psgneldm2 17747 psgnfitr 17760 iscyggen2 18106 iscyg3 18111 cycsubgcyg 18125 eldprd 18226 leordtval2 20826 iocpnfordt 20829 icomnfordt 20830 lecldbas 20833 tsmsxplem1 21766 minveclem2 23005 lhop2 23582 taylthlem2 23932 fsumvma 24738 dchrptlem2 24790 2sqlem1 24942 dchrisum0fno1 25000 minvecolem2 27115 gsumesum 29448 esumlub 29449 esumcst 29452 esumpcvgval 29467 esumgect 29479 esum2d 29482 sigapildsys 29552 sxbrsigalem2 29675 omssubaddlem 29688 omssubadd 29689 eulerpartgbij 29761 bnj1366 30154 msubco 30682 msubvrs 30711 fin2so 32566 poimirlem17 32596 poimirlem20 32599 cntotbnd 32765 islsat 33296 |
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