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Mirrors > Home > MPE Home > Th. List > nfmpo2 | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.) |
Ref | Expression |
---|---|
nfmpo2 | ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mpo 7161 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} | |
2 | nfoprab2 7216 | . 2 ⊢ Ⅎ𝑦{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} | |
3 | 1, 2 | nfcxfr 2975 | 1 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 Ⅎwnfc 2961 {coprab 7157 ∈ cmpo 7158 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-oprab 7160 df-mpo 7161 |
This theorem is referenced by: ovmpos 7298 ov2gf 7299 ovmpodxf 7300 ovmpodf 7306 ovmpodv2 7308 xpcomco 8607 mapxpen 8683 pwfseqlem2 10081 pwfseqlem4a 10083 pwfseqlem4 10084 gsum2d2lem 19093 gsum2d2 19094 gsumcom2 19095 dprd2d2 19166 cnmpt21 22279 cnmpt2t 22281 cnmptcom 22286 cnmpt2k 22296 xkocnv 22422 finxpreclem2 34674 finxpreclem6 34680 fmuldfeq 41913 smflimlem6 43101 ovmpordxf 44436 |
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