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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 7161 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑤〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)} | |
2 | nfmpo.1 | . . . . . 6 ⊢ Ⅎ𝑧𝐴 | |
3 | 2 | nfcri 2971 | . . . . 5 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴 |
4 | nfmpo.2 | . . . . . 6 ⊢ Ⅎ𝑧𝐵 | |
5 | 4 | nfcri 2971 | . . . . 5 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐵 |
6 | 3, 5 | nfan 1900 | . . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) |
7 | nfmpo.3 | . . . . 5 ⊢ Ⅎ𝑧𝐶 | |
8 | 7 | nfeq2 2995 | . . . 4 ⊢ Ⅎ𝑧 𝑤 = 𝐶 |
9 | 6, 8 | nfan 1900 | . . 3 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶) |
10 | 9 | nfoprab 7218 | . 2 ⊢ Ⅎ𝑧{〈〈𝑥, 𝑦〉, 𝑤〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)} |
11 | 1, 10 | nfcxfr 2975 | 1 ⊢ Ⅎ𝑧(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-tru 1540 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: nfof 7414 el2mpocsbcl 7780 nfseq 13380 ptbasfi 22189 sdclem1 35033 fmuldfeqlem1 41912 stoweidlem51 42385 vonicc 43016 |
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