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Mirrors > Home > ILE Home > Th. List > nfco | GIF version |
Description: Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.) |
Ref | Expression |
---|---|
nfco.1 | ⊢ Ⅎ𝑥𝐴 |
nfco.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfco | ⊢ Ⅎ𝑥(𝐴 ∘ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-co 4400 | . 2 ⊢ (𝐴 ∘ 𝐵) = {〈𝑦, 𝑧〉 ∣ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)} | |
2 | nfcv 2223 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
3 | nfco.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
4 | nfcv 2223 | . . . . . 6 ⊢ Ⅎ𝑥𝑤 | |
5 | 2, 3, 4 | nfbr 3849 | . . . . 5 ⊢ Ⅎ𝑥 𝑦𝐵𝑤 |
6 | nfco.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
7 | nfcv 2223 | . . . . . 6 ⊢ Ⅎ𝑥𝑧 | |
8 | 4, 6, 7 | nfbr 3849 | . . . . 5 ⊢ Ⅎ𝑥 𝑤𝐴𝑧 |
9 | 5, 8 | nfan 1498 | . . . 4 ⊢ Ⅎ𝑥(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧) |
10 | 9 | nfex 1569 | . . 3 ⊢ Ⅎ𝑥∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧) |
11 | 10 | nfopab 3866 | . 2 ⊢ Ⅎ𝑥{〈𝑦, 𝑧〉 ∣ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)} |
12 | 1, 11 | nfcxfr 2220 | 1 ⊢ Ⅎ𝑥(𝐴 ∘ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ∃wex 1422 Ⅎwnfc 2210 class class class wbr 3805 {copab 3858 ∘ ccom 4395 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-v 2612 df-un 2986 df-sn 3422 df-pr 3423 df-op 3425 df-br 3806 df-opab 3860 df-co 4400 |
This theorem is referenced by: nffun 4974 nftpos 5948 |
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