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Mirrors > Home > MPE Home > Th. List > fveqeq2d | Structured version Visualization version GIF version |
Description: Equality deduction for function value. (Contributed by BJ, 30-Aug-2022.) |
Ref | Expression |
---|---|
fveqeq2d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
fveqeq2d | ⊢ (𝜑 → ((𝐹‘𝐴) = 𝐶 ↔ (𝐹‘𝐵) = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveqeq2d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | fveq2d 6910 | . 2 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐹‘𝐵)) |
3 | 2 | eqeq1d 2736 | 1 ⊢ (𝜑 → ((𝐹‘𝐴) = 𝐶 ↔ (𝐹‘𝐵) = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1536 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-iota 6515 df-fv 6570 |
This theorem is referenced by: fveqeq2 6915 op1stg 8024 op2ndg 8025 ttrclss 9757 ttrclselem2 9763 fpwwecbv 10681 fpwwelem 10682 fseq1m1p1 13635 ico01fl0 13855 divfl0 13860 hashssdif 14447 cshw1 14856 smumullem 16525 algcvga 16612 vdwlem6 17019 vdwlem8 17021 ramub1lem1 17059 resmgmhm 18736 resmhm 18845 isghmOLD 19246 fislw 19657 pgpfaclem2 20116 0ringdif 20543 abvfval 20827 abvpropd 20852 lspsneq0 21027 reslmhm 21068 lspsneq 21141 mdetunilem7 22639 imasdsf1olem 24398 bcth 25376 ovoliunnul 25555 lognegb 26646 vmaval 27170 2lgslem3c 27456 2lgslem3d 27457 rusgrnumwrdl2 29618 wlkiswwlks2 29904 rusgrnumwwlks 30003 clwlkclwwlklem1 30027 clwlkclwwlklem2 30028 numclwwlk1 30389 wlkl0 30395 numclwlk1lem1 30397 isnvlem 30638 lnoval 30780 normsub0 31164 elunop2 32041 ishst 32242 hstri 32293 aciunf1lem 32678 lmatfval 33774 lmatcl 33776 voliune 34209 volfiniune 34210 snmlval 35315 voliunnfl 37650 sdclem1 37729 islshp 38960 lshpnel2N 38966 lshpset2N 39100 dicffval 41156 dicfval 41157 mapdhval 41706 hdmap1fval 41778 hdmap1vallem 41779 hdmap1val 41780 aks6d1c6isolem1 42155 aks6d1c6lem5 42158 diophin 42759 eldioph4b 42798 eldioph4i 42799 diophren 42800 fperiodmullem 45253 fargshiftfva 47367 paireqne 47435 isuspgrim0 47809 grimidvtxedg 47813 grimcnv 47816 grimco 47817 uhgrimisgrgriclem 47835 clnbgrgrimlem 47838 |
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