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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 7666 | . 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 7435 | . . . 4 ⊢ Ⅎ𝑥((𝑢‘𝑤)𝑅(𝑣‘𝑤)) |
8 | 3, 7 | nfmpt 5254 | . . 3 ⊢ Ⅎ𝑥(𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤))) |
9 | 2, 2, 8 | nfmpo 7487 | . 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 2883 Vcvv 3474 ∩ cin 3946 ↦ cmpt 5230 dom cdm 5675 ‘cfv 6540 (class class class)co 7405 ∈ cmpo 7407 ∘f cof 7664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-iota 6492 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 |
This theorem is referenced by: (None) |
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