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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 6044 | . 2 ⊢ ∘𝑓 𝑅 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤)))) | |
2 | nfcv 2306 | . . 3 ⊢ Ⅎ𝑥V | |
3 | nfcv 2306 | . . . 4 ⊢ Ⅎ𝑥(dom 𝑢 ∩ dom 𝑣) | |
4 | nfcv 2306 | . . . . 5 ⊢ Ⅎ𝑥(𝑢‘𝑤) | |
5 | nfof.1 | . . . . 5 ⊢ Ⅎ𝑥𝑅 | |
6 | nfcv 2306 | . . . . 5 ⊢ Ⅎ𝑥(𝑣‘𝑤) | |
7 | 4, 5, 6 | nfov 5863 | . . . 4 ⊢ Ⅎ𝑥((𝑢‘𝑤)𝑅(𝑣‘𝑤)) |
8 | 3, 7 | nfmpt 4068 | . . 3 ⊢ Ⅎ𝑥(𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤))) |
9 | 2, 2, 8 | nfmpo 5902 | . 2 ⊢ Ⅎ𝑥(𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤)))) |
10 | 1, 9 | nfcxfr 2303 | 1 ⊢ Ⅎ𝑥 ∘𝑓 𝑅 |
Colors of variables: wff set class |
Syntax hints: Ⅎwnfc 2293 Vcvv 2721 ∩ cin 3110 ↦ cmpt 4037 dom cdm 4598 ‘cfv 5182 (class class class)co 5836 ∈ cmpo 5838 ∘𝑓 cof 6042 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-v 2723 df-un 3115 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-iota 5147 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-of 6044 |
This theorem is referenced by: (None) |
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