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Mirrors > Home > MPE Home > Th. List > nfco | Structured version Visualization version 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 5047 | . 2 ⊢ (𝐴 ∘ 𝐵) = {〈𝑦, 𝑧〉 ∣ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)} | |
2 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
3 | nfco.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
4 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑥𝑤 | |
5 | 2, 3, 4 | nfbr 4629 | . . . . 5 ⊢ Ⅎ𝑥 𝑦𝐵𝑤 |
6 | nfco.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
7 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑥𝑧 | |
8 | 4, 6, 7 | nfbr 4629 | . . . . 5 ⊢ Ⅎ𝑥 𝑤𝐴𝑧 |
9 | 5, 8 | nfan 1816 | . . . 4 ⊢ Ⅎ𝑥(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧) |
10 | 9 | nfex 2140 | . . 3 ⊢ Ⅎ𝑥∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧) |
11 | 10 | nfopab 4650 | . 2 ⊢ Ⅎ𝑥{〈𝑦, 𝑧〉 ∣ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)} |
12 | 1, 11 | nfcxfr 2749 | 1 ⊢ Ⅎ𝑥(𝐴 ∘ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∃wex 1695 Ⅎwnfc 2738 class class class wbr 4583 {copab 4642 ∘ ccom 5042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-co 5047 |
This theorem is referenced by: nffun 5826 nftpos 7274 cnmpt11 21276 cnmpt21 21284 poimirlem16 32595 poimirlem19 32598 csbcog 36960 choicefi 38387 cncficcgt0 38774 volioofmpt 38887 volicofmpt 38890 stoweidlem31 38924 stoweidlem59 38952 |
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