Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 7398 | . 2 ⊢ ∘f 𝑅 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤)))) | |
2 | nfcv 2974 | . . 3 ⊢ Ⅎ𝑥V | |
3 | nfcv 2974 | . . . 4 ⊢ Ⅎ𝑥(dom 𝑢 ∩ dom 𝑣) | |
4 | nfcv 2974 | . . . . 5 ⊢ Ⅎ𝑥(𝑢‘𝑤) | |
5 | nfof.1 | . . . . 5 ⊢ Ⅎ𝑥𝑅 | |
6 | nfcv 2974 | . . . . 5 ⊢ Ⅎ𝑥(𝑣‘𝑤) | |
7 | 4, 5, 6 | nfov 7175 | . . . 4 ⊢ Ⅎ𝑥((𝑢‘𝑤)𝑅(𝑣‘𝑤)) |
8 | 3, 7 | nfmpt 5154 | . . 3 ⊢ Ⅎ𝑥(𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤))) |
9 | 2, 2, 8 | nfmpo 7225 | . 2 ⊢ Ⅎ𝑥(𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤)))) |
10 | 1, 9 | nfcxfr 2972 | 1 ⊢ Ⅎ𝑥 ∘f 𝑅 |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2958 Vcvv 3492 ∩ cin 3932 ↦ cmpt 5137 dom cdm 5548 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 ∘f cof 7396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-iota 6307 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |