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Mirrors > Home > MPE Home > Th. List > fvconst2g | Structured version Visualization version GIF version |
Description: The value of a constant function. (Contributed by NM, 20-Aug-2005.) |
Ref | Expression |
---|---|
fvconst2g | ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝐶) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fconstg 6329 | . 2 ⊢ (𝐵 ∈ 𝐷 → (𝐴 × {𝐵}):𝐴⟶{𝐵}) | |
2 | fvconst 6682 | . 2 ⊢ (((𝐴 × {𝐵}):𝐴⟶{𝐵} ∧ 𝐶 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝐶) = 𝐵) | |
3 | 1, 2 | sylan 577 | 1 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝐶) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 {csn 4397 × cxp 5340 ⟶wf 6119 ‘cfv 6123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-fv 6131 |
This theorem is referenced by: fconst2g 6724 fvconst2 6725 ofc1 7180 ofc2 7181 caofid0l 7185 caofid0r 7186 caofid1 7187 caofid2 7188 fnsuppres 7587 ser0 13147 ser1const 13151 exp1 13160 expp1 13161 climconst2 14656 climaddc1 14742 climmulc2 14744 climsubc1 14745 climsubc2 14746 climlec2 14766 fsumconst 14896 supcvg 14962 prodf1 14996 prod0 15046 fprodconst 15081 seq1st 15657 algr0 15658 algrf 15659 ramz 16100 pwsbas 16500 pwsplusgval 16503 pwsmulrval 16504 pwsle 16505 pwsvscafval 16507 pwspjmhm 17721 pwsco1mhm 17723 pwsinvg 17882 mulg1 17902 mulgnnp1 17903 mulgnnsubcl 17907 mulgnn0z 17920 mulgnndir 17922 mulgnn0di 18584 gsumconst 18687 pwslmod 19329 psrlidm 19764 coe1tm 20003 coe1fzgsumd 20032 evl1scad 20059 frlmvscaval 20474 decpmatid 20945 pmatcollpwscmatlem1 20964 lmconst 21436 cnconst2 21458 xkoptsub 21828 xkopt 21829 xkopjcn 21830 tmdgsum 22269 tmdgsum2 22270 symgtgp 22275 cstucnd 22458 pcoptcl 23190 pcopt 23191 pcopt2 23192 dvidlem 24078 dvconst 24079 dvnff 24085 dvn0 24086 dvcmul 24106 dvcmulf 24107 fta1blem 24327 plyeq0lem 24365 coemulc 24410 dgreq0 24420 dgrmulc 24426 qaa 24477 dchrisumlema 25590 ofcc 30713 ofcof 30714 sseqf 31000 sseqp1 31003 cvmlift3lem9 31855 ismrer1 34179 dvsinax 40922 stoweidlem21 41032 stoweidlem47 41058 elaa2 41245 zlmodzxzscm 42982 2sphere0 43302 |
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