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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 4653 | . 2 ⊢ (𝐴 ∘ 𝐵) = {〈𝑦, 𝑧〉 ∣ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)} | |
2 | nfcv 2332 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
3 | nfco.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
4 | nfcv 2332 | . . . . . 6 ⊢ Ⅎ𝑥𝑤 | |
5 | 2, 3, 4 | nfbr 4064 | . . . . 5 ⊢ Ⅎ𝑥 𝑦𝐵𝑤 |
6 | nfco.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
7 | nfcv 2332 | . . . . . 6 ⊢ Ⅎ𝑥𝑧 | |
8 | 4, 6, 7 | nfbr 4064 | . . . . 5 ⊢ Ⅎ𝑥 𝑤𝐴𝑧 |
9 | 5, 8 | nfan 1576 | . . . 4 ⊢ Ⅎ𝑥(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧) |
10 | 9 | nfex 1648 | . . 3 ⊢ Ⅎ𝑥∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧) |
11 | 10 | nfopab 4086 | . 2 ⊢ Ⅎ𝑥{〈𝑦, 𝑧〉 ∣ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)} |
12 | 1, 11 | nfcxfr 2329 | 1 ⊢ Ⅎ𝑥(𝐴 ∘ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∃wex 1503 Ⅎwnfc 2319 class class class wbr 4018 {copab 4078 ∘ ccom 4648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-un 3148 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-co 4653 |
This theorem is referenced by: nffun 5258 nftpos 6303 cnmpt11 14235 cnmpt21 14243 |
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