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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 5678 | . 2 ⊢ (𝐴 ∘ 𝐵) = {〈𝑦, 𝑧〉 ∣ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)} | |
2 | nfcv 2902 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
3 | nfco.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
4 | nfcv 2902 | . . . . . 6 ⊢ Ⅎ𝑥𝑤 | |
5 | 2, 3, 4 | nfbr 5188 | . . . . 5 ⊢ Ⅎ𝑥 𝑦𝐵𝑤 |
6 | nfco.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
7 | nfcv 2902 | . . . . . 6 ⊢ Ⅎ𝑥𝑧 | |
8 | 4, 6, 7 | nfbr 5188 | . . . . 5 ⊢ Ⅎ𝑥 𝑤𝐴𝑧 |
9 | 5, 8 | nfan 1902 | . . . 4 ⊢ Ⅎ𝑥(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧) |
10 | 9 | nfex 2317 | . . 3 ⊢ Ⅎ𝑥∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧) |
11 | 10 | nfopab 5210 | . 2 ⊢ Ⅎ𝑥{〈𝑦, 𝑧〉 ∣ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)} |
12 | 1, 11 | nfcxfr 2900 | 1 ⊢ Ⅎ𝑥(𝐴 ∘ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∃wex 1781 Ⅎwnfc 2882 class class class wbr 5141 {copab 5203 ∘ ccom 5673 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-co 5678 |
This theorem is referenced by: csbcog 6285 nffun 6560 nftpos 8228 nfwrecs 8283 cnmpt11 23096 cnmpt21 23104 poimirlem16 36308 poimirlem19 36311 choicefi 43670 cncficcgt0 44377 volioofmpt 44483 volicofmpt 44486 stoweidlem31 44520 stoweidlem59 44548 |
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