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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 5915 | . 2 ⊢ ∘𝑟 𝑅 = {〈𝑢, 𝑣〉 ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)} | |
2 | nfcv 2240 | . . . 4 ⊢ Ⅎ𝑥(dom 𝑢 ∩ dom 𝑣) | |
3 | nfcv 2240 | . . . . 5 ⊢ Ⅎ𝑥(𝑢‘𝑤) | |
4 | nfof.1 | . . . . 5 ⊢ Ⅎ𝑥𝑅 | |
5 | nfcv 2240 | . . . . 5 ⊢ Ⅎ𝑥(𝑣‘𝑤) | |
6 | 3, 4, 5 | nfbr 3919 | . . . 4 ⊢ Ⅎ𝑥(𝑢‘𝑤)𝑅(𝑣‘𝑤) |
7 | 2, 6 | nfralxy 2430 | . . 3 ⊢ Ⅎ𝑥∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤) |
8 | 7 | nfopab 3936 | . 2 ⊢ Ⅎ𝑥{〈𝑢, 𝑣〉 ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)} |
9 | 1, 8 | nfcxfr 2237 | 1 ⊢ Ⅎ𝑥 ∘𝑟 𝑅 |
Colors of variables: wff set class |
Syntax hints: Ⅎwnfc 2227 ∀wral 2375 ∩ cin 3020 class class class wbr 3875 {copab 3928 dom cdm 4477 ‘cfv 5059 ∘𝑟 cofr 5913 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-v 2643 df-un 3025 df-sn 3480 df-pr 3481 df-op 3483 df-br 3876 df-opab 3930 df-ofr 5915 |
This theorem is referenced by: (None) |
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