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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 7623 | . 2 ⊢ ∘r 𝑅 = {⟨𝑢, 𝑣⟩ ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)} | |
2 | nfcv 2908 | . . . 4 ⊢ Ⅎ𝑥(dom 𝑢 ∩ dom 𝑣) | |
3 | nfcv 2908 | . . . . 5 ⊢ Ⅎ𝑥(𝑢‘𝑤) | |
4 | nfof.1 | . . . . 5 ⊢ Ⅎ𝑥𝑅 | |
5 | nfcv 2908 | . . . . 5 ⊢ Ⅎ𝑥(𝑣‘𝑤) | |
6 | 3, 4, 5 | nfbr 5157 | . . . 4 ⊢ Ⅎ𝑥(𝑢‘𝑤)𝑅(𝑣‘𝑤) |
7 | 2, 6 | nfralw 3297 | . . 3 ⊢ Ⅎ𝑥∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤) |
8 | 7 | nfopab 5179 | . 2 ⊢ Ⅎ𝑥{⟨𝑢, 𝑣⟩ ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)} |
9 | 1, 8 | nfcxfr 2906 | 1 ⊢ Ⅎ𝑥 ∘r 𝑅 |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2888 ∀wral 3065 ∩ cin 3914 class class class wbr 5110 {copab 5172 dom cdm 5638 ‘cfv 6501 ∘r cofr 7621 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-ofr 7623 |
This theorem is referenced by: (None) |
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