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Mirrors > Home > MPE Home > Th. List > imaeq1i | Structured version Visualization version GIF version |
Description: Equality theorem for image. (Contributed by NM, 21-Dec-2008.) |
Ref | Expression |
---|---|
imaeq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
imaeq1i | ⊢ (𝐴 “ 𝐶) = (𝐵 “ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imaeq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | imaeq1 5380 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 “ 𝐶) = (𝐵 “ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 “ cima 5041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: mptpreima 5545 isarep2 5892 suppun 7202 supp0cosupp0 7221 imacosupp 7222 fsuppun 8177 fsuppcolem 8189 marypha2lem4 8227 dfoi 8299 r1limg 8517 isf34lem3 9080 compss 9081 fpwwe2lem13 9343 infrenegsup 10883 gsumzf1o 18136 ssidcn 20869 cnco 20880 qtopres 21311 idqtop 21319 qtopcn 21327 mbfid 23209 mbfres 23217 cncombf 23231 dvlog 24197 efopnlem2 24203 disjpreima 28779 imadifxp 28796 rinvf1o 28814 mbfmcst 29648 mbfmco 29653 sitmcl 29740 eulerpartlemt 29760 eulerpartlemmf 29764 eulerpart 29771 0rrv 29840 mclsppslem 30734 csbpredg 32348 mptsnun 32362 poimirlem3 32582 ftc1anclem3 32657 areacirclem5 32674 cytpval 36806 arearect 36820 brtrclfv2 37038 0cnf 38762 mbf0 38849 fourierdlem62 39061 smfco 39687 eucrct2eupth 41413 |
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