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Mirrors > Home > ILE Home > Th. List > nfofr | GIF version |
Description: Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.) |
Ref | Expression |
---|---|
nfof.1 | ⊢ Ⅎ𝑥𝑅 |
Ref | Expression |
---|---|
nfofr | ⊢ Ⅎ𝑥 ∘𝑟 𝑅 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ofr 5951 | . 2 ⊢ ∘𝑟 𝑅 = {〈𝑢, 𝑣〉 ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)} | |
2 | nfcv 2258 | . . . 4 ⊢ Ⅎ𝑥(dom 𝑢 ∩ dom 𝑣) | |
3 | nfcv 2258 | . . . . 5 ⊢ Ⅎ𝑥(𝑢‘𝑤) | |
4 | nfof.1 | . . . . 5 ⊢ Ⅎ𝑥𝑅 | |
5 | nfcv 2258 | . . . . 5 ⊢ Ⅎ𝑥(𝑣‘𝑤) | |
6 | 3, 4, 5 | nfbr 3944 | . . . 4 ⊢ Ⅎ𝑥(𝑢‘𝑤)𝑅(𝑣‘𝑤) |
7 | 2, 6 | nfralxy 2448 | . . 3 ⊢ Ⅎ𝑥∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤) |
8 | 7 | nfopab 3966 | . 2 ⊢ Ⅎ𝑥{〈𝑢, 𝑣〉 ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)} |
9 | 1, 8 | nfcxfr 2255 | 1 ⊢ Ⅎ𝑥 ∘𝑟 𝑅 |
Colors of variables: wff set class |
Syntax hints: Ⅎwnfc 2245 ∀wral 2393 ∩ cin 3040 class class class wbr 3899 {copab 3958 dom cdm 4509 ‘cfv 5093 ∘𝑟 cofr 5949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-v 2662 df-un 3045 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-ofr 5951 |
This theorem is referenced by: (None) |
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