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Mirrors > Home > MPE Home > Th. List > nfmpo | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.) |
Ref | Expression |
---|---|
nfmpo.1 | ⊢ Ⅎ𝑧𝐴 |
nfmpo.2 | ⊢ Ⅎ𝑧𝐵 |
nfmpo.3 | ⊢ Ⅎ𝑧𝐶 |
Ref | Expression |
---|---|
nfmpo | ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mpo 7414 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)} | |
2 | nfmpo.1 | . . . . . 6 ⊢ Ⅎ𝑧𝐴 | |
3 | 2 | nfcri 2891 | . . . . 5 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴 |
4 | nfmpo.2 | . . . . . 6 ⊢ Ⅎ𝑧𝐵 | |
5 | 4 | nfcri 2891 | . . . . 5 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐵 |
6 | 3, 5 | nfan 1903 | . . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) |
7 | nfmpo.3 | . . . . 5 ⊢ Ⅎ𝑧𝐶 | |
8 | 7 | nfeq2 2921 | . . . 4 ⊢ Ⅎ𝑧 𝑤 = 𝐶 |
9 | 6, 8 | nfan 1903 | . . 3 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶) |
10 | 9 | nfoprab 7473 | . 2 ⊢ Ⅎ𝑧{⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)} |
11 | 1, 10 | nfcxfr 2902 | 1 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 Ⅎwnfc 2884 {coprab 7410 ∈ cmpo 7411 |
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 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-oprab 7413 df-mpo 7414 |
This theorem is referenced by: nfof 7676 el2mpocsbcl 8071 nfseq 13976 ptbasfi 23085 sdclem1 36611 fmuldfeqlem1 44298 stoweidlem51 44767 vonicc 45401 |
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