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Mirrors > Home > MPE Home > Th. List > imaeq1 | Structured version Visualization version GIF version |
Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.) |
Ref | Expression |
---|---|
imaeq1 | ⊢ (𝐴 = 𝐵 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reseq1 5311 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)) | |
2 | 1 | rneqd 5274 | . 2 ⊢ (𝐴 = 𝐵 → ran (𝐴 ↾ 𝐶) = ran (𝐵 ↾ 𝐶)) |
3 | df-ima 5051 | . 2 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶) | |
4 | df-ima 5051 | . 2 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶) | |
5 | 2, 3, 4 | 3eqtr4g 2669 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ran crn 5039 ↾ cres 5040 “ 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: imaeq1i 5382 imaeq1d 5384 suppval 7184 eceq2 7671 marypha1lem 8222 marypha1 8223 ackbij2lem2 8945 ackbij2lem3 8946 r1om 8949 limsupval 14053 isacs1i 16141 mreacs 16142 islindf 19970 iscnp 20851 xkoccn 21232 xkohaus 21266 xkoco1cn 21270 xkoco2cn 21271 xkococnlem 21272 xkococn 21273 xkoinjcn 21300 fmval 21557 fmf 21559 utoptop 21848 restutop 21851 restutopopn 21852 ustuqtoplem 21853 ustuqtop1 21855 ustuqtop2 21856 ustuqtop4 21858 ustuqtop5 21859 utopsnneiplem 21861 utopsnnei 21863 neipcfilu 21910 psmetutop 22182 cfilfval 22870 elply2 23756 coeeu 23785 coelem 23786 coeeq 23787 dmarea 24484 mclsax 30720 tailfval 31537 bj-cleq 32142 poimirlem15 32594 poimirlem24 32603 brtrclfv2 37038 |
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