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Mirrors > Home > MPE Home > Th. List > nfof | Structured version Visualization version 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 7697 | . 2 ⊢ ∘f 𝑅 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤)))) | |
2 | nfcv 2903 | . . 3 ⊢ Ⅎ𝑥V | |
3 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑥(dom 𝑢 ∩ dom 𝑣) | |
4 | nfcv 2903 | . . . . 5 ⊢ Ⅎ𝑥(𝑢‘𝑤) | |
5 | nfof.1 | . . . . 5 ⊢ Ⅎ𝑥𝑅 | |
6 | nfcv 2903 | . . . . 5 ⊢ Ⅎ𝑥(𝑣‘𝑤) | |
7 | 4, 5, 6 | nfov 7461 | . . . 4 ⊢ Ⅎ𝑥((𝑢‘𝑤)𝑅(𝑣‘𝑤)) |
8 | 3, 7 | nfmpt 5255 | . . 3 ⊢ Ⅎ𝑥(𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤))) |
9 | 2, 2, 8 | nfmpo 7515 | . 2 ⊢ Ⅎ𝑥(𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤)))) |
10 | 1, 9 | nfcxfr 2901 | 1 ⊢ Ⅎ𝑥 ∘f 𝑅 |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2888 Vcvv 3478 ∩ cin 3962 ↦ cmpt 5231 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ∘f cof 7695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-iota 6516 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 |
This theorem is referenced by: (None) |
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