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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 7416 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} | |
2 | nfoprab2 7473 | . 2 ⊢ Ⅎ𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} | |
3 | 1, 2 | nfcxfr 2901 | 1 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 Ⅎwnfc 2883 {coprab 7412 ∈ cmpo 7413 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-oprab 7415 df-mpo 7416 |
This theorem is referenced by: ovmpos 7558 ov2gf 7559 ovmpodxf 7560 ovmpodf 7566 ovmpodv2 7568 xpcomco 9064 mapxpen 9145 pwfseqlem2 10656 pwfseqlem4a 10658 pwfseqlem4 10659 gsum2d2lem 19882 gsum2d2 19883 gsumcom2 19884 dprd2d2 19955 cnmpt21 23395 cnmpt2t 23397 cnmptcom 23402 cnmpt2k 23412 xkocnv 23538 finxpreclem2 36574 finxpreclem6 36580 mnringmulrcld 43289 fmuldfeq 44598 smflimlem6 45791 ovmpordxf 47103 |
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