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Mirrors > Home > MPE Home > Th. List > Mathboxes > ofsubid | Structured version Visualization version GIF version |
Description: Function analogue of subid 11515. (Contributed by Steve Rodriguez, 5-Nov-2015.) |
Ref | Expression |
---|---|
ofsubid | β’ ((π΄ β π β§ πΉ:π΄βΆβ) β (πΉ βf β πΉ) = (π΄ Γ {0})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 481 | . 2 β’ ((π΄ β π β§ πΉ:π΄βΆβ) β π΄ β π) | |
2 | ffn 6725 | . . 3 β’ (πΉ:π΄βΆβ β πΉ Fn π΄) | |
3 | 2 | adantl 480 | . 2 β’ ((π΄ β π β§ πΉ:π΄βΆβ) β πΉ Fn π΄) |
4 | c0ex 11244 | . . . 4 β’ 0 β V | |
5 | 4 | fconst 6786 | . . 3 β’ (π΄ Γ {0}):π΄βΆ{0} |
6 | ffn 6725 | . . 3 β’ ((π΄ Γ {0}):π΄βΆ{0} β (π΄ Γ {0}) Fn π΄) | |
7 | 5, 6 | mp1i 13 | . 2 β’ ((π΄ β π β§ πΉ:π΄βΆβ) β (π΄ Γ {0}) Fn π΄) |
8 | eqidd 2728 | . 2 β’ (((π΄ β π β§ πΉ:π΄βΆβ) β§ π₯ β π΄) β (πΉβπ₯) = (πΉβπ₯)) | |
9 | ffvelcdm 7094 | . . . . 5 β’ ((πΉ:π΄βΆβ β§ π₯ β π΄) β (πΉβπ₯) β β) | |
10 | 9 | subidd 11595 | . . . 4 β’ ((πΉ:π΄βΆβ β§ π₯ β π΄) β ((πΉβπ₯) β (πΉβπ₯)) = 0) |
11 | 10 | adantll 712 | . . 3 β’ (((π΄ β π β§ πΉ:π΄βΆβ) β§ π₯ β π΄) β ((πΉβπ₯) β (πΉβπ₯)) = 0) |
12 | 4 | fvconst2 7220 | . . . 4 β’ (π₯ β π΄ β ((π΄ Γ {0})βπ₯) = 0) |
13 | 12 | adantl 480 | . . 3 β’ (((π΄ β π β§ πΉ:π΄βΆβ) β§ π₯ β π΄) β ((π΄ Γ {0})βπ₯) = 0) |
14 | 11, 13 | eqtr4d 2770 | . 2 β’ (((π΄ β π β§ πΉ:π΄βΆβ) β§ π₯ β π΄) β ((πΉβπ₯) β (πΉβπ₯)) = ((π΄ Γ {0})βπ₯)) |
15 | 1, 3, 3, 7, 8, 8, 14 | offveq 7713 | 1 β’ ((π΄ β π β§ πΉ:π΄βΆβ) β (πΉ βf β πΉ) = (π΄ Γ {0})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 {csn 4630 Γ cxp 5678 Fn wfn 6546 βΆwf 6547 βcfv 6551 (class class class)co 7424 βf cof 7687 βcc 11142 0cc0 11144 β cmin 11480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-po 5592 df-so 5593 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7689 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-ltxr 11289 df-sub 11482 |
This theorem is referenced by: expgrowth 43775 |
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