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Mirrors > Home > MPE Home > Th. List > nfofr | Structured version Visualization version GIF version |
Description: Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.) |
Ref | Expression |
---|---|
nfof.1 | ⊢ Ⅎ𝑥𝑅 |
Ref | Expression |
---|---|
nfofr | ⊢ Ⅎ𝑥 ∘r 𝑅 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ofr 7410 | . 2 ⊢ ∘r 𝑅 = {〈𝑢, 𝑣〉 ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)} | |
2 | nfcv 2977 | . . . 4 ⊢ Ⅎ𝑥(dom 𝑢 ∩ dom 𝑣) | |
3 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑥(𝑢‘𝑤) | |
4 | nfof.1 | . . . . 5 ⊢ Ⅎ𝑥𝑅 | |
5 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑥(𝑣‘𝑤) | |
6 | 3, 4, 5 | nfbr 5113 | . . . 4 ⊢ Ⅎ𝑥(𝑢‘𝑤)𝑅(𝑣‘𝑤) |
7 | 2, 6 | nfralw 3225 | . . 3 ⊢ Ⅎ𝑥∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤) |
8 | 7 | nfopab 5134 | . 2 ⊢ Ⅎ𝑥{〈𝑢, 𝑣〉 ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)} |
9 | 1, 8 | nfcxfr 2975 | 1 ⊢ Ⅎ𝑥 ∘r 𝑅 |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2961 ∀wral 3138 ∩ cin 3935 class class class wbr 5066 {copab 5128 dom cdm 5555 ‘cfv 6355 ∘r cofr 7408 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-ofr 7410 |
This theorem is referenced by: (None) |
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