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Mirrors > Home > ILE Home > Th. List > nfof | GIF version |
Description: Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.) |
Ref | Expression |
---|---|
nfof.1 | ⊢ Ⅎ𝑥𝑅 |
Ref | Expression |
---|---|
nfof | ⊢ Ⅎ𝑥 ∘𝑓 𝑅 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-of 5982 | . 2 ⊢ ∘𝑓 𝑅 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤)))) | |
2 | nfcv 2281 | . . 3 ⊢ Ⅎ𝑥V | |
3 | nfcv 2281 | . . . 4 ⊢ Ⅎ𝑥(dom 𝑢 ∩ dom 𝑣) | |
4 | nfcv 2281 | . . . . 5 ⊢ Ⅎ𝑥(𝑢‘𝑤) | |
5 | nfof.1 | . . . . 5 ⊢ Ⅎ𝑥𝑅 | |
6 | nfcv 2281 | . . . . 5 ⊢ Ⅎ𝑥(𝑣‘𝑤) | |
7 | 4, 5, 6 | nfov 5801 | . . . 4 ⊢ Ⅎ𝑥((𝑢‘𝑤)𝑅(𝑣‘𝑤)) |
8 | 3, 7 | nfmpt 4020 | . . 3 ⊢ Ⅎ𝑥(𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤))) |
9 | 2, 2, 8 | nfmpo 5840 | . 2 ⊢ Ⅎ𝑥(𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤)))) |
10 | 1, 9 | nfcxfr 2278 | 1 ⊢ Ⅎ𝑥 ∘𝑓 𝑅 |
Colors of variables: wff set class |
Syntax hints: Ⅎwnfc 2268 Vcvv 2686 ∩ cin 3070 ↦ cmpt 3989 dom cdm 4539 ‘cfv 5123 (class class class)co 5774 ∈ cmpo 5776 ∘𝑓 cof 5980 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-iota 5088 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-of 5982 |
This theorem is referenced by: (None) |
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