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Mirrors > Home > MPE Home > Th. List > rneq | Structured version Visualization version GIF version |
Description: Equality theorem for range. (Contributed by NM, 29-Dec-1996.) |
Ref | Expression |
---|---|
rneq | ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnveq 5296 | . . 3 ⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵) | |
2 | 1 | dmeqd 5326 | . 2 ⊢ (𝐴 = 𝐵 → dom ◡𝐴 = dom ◡𝐵) |
3 | df-rn 5125 | . 2 ⊢ ran 𝐴 = dom ◡𝐴 | |
4 | df-rn 5125 | . 2 ⊢ ran 𝐵 = dom ◡𝐵 | |
5 | 2, 3, 4 | 3eqtr4g 2681 | 1 ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ◡ccnv 5113 dom cdm 5114 ran crn 5115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-cnv 5122 df-dm 5124 df-rn 5125 |
This theorem is referenced by: rneqi 5352 rneqd 5353 feq1 6026 foeq1 6111 fnrnfv 6242 fconst5 6471 frxp 7287 tz7.44-2 7503 tz7.44-3 7504 ixpsnf1o 7948 ordtypecbv 8422 ordtypelem3 8425 dfac8alem 8852 dfac8a 8853 dfac5lem3 8948 dfac9 8958 dfac12lem1 8965 dfac12r 8968 ackbij2 9065 isfin3ds 9151 fin23lem17 9160 fin23lem29 9163 fin23lem30 9164 fin23lem32 9166 fin23lem34 9168 fin23lem35 9169 fin23lem39 9172 fin23lem41 9174 isf33lem 9188 isf34lem6 9202 dcomex 9269 axdc2lem 9270 zorn2lem1 9318 zorn2g 9325 ttukey2g 9338 gruurn 9620 rpnnen1lem6 11819 rpnnen1OLD 11825 relexp0g 13762 relexpsucnnr 13765 dfrtrcl2 13802 mpfrcl 19518 ply1frcl 19683 pnrmopn 21147 isi1f 23441 itg1val 23450 axlowdimlem13 25834 axlowdim1 25839 ausgrusgri 26063 0uhgrsubgr 26171 cusgrsize 26350 ex-rn 27297 gidval 27366 grpoinvfval 27376 grpodivfval 27388 isablo 27400 vciOLD 27416 isvclem 27432 isnvlem 27465 isphg 27672 pj11i 28570 hmopidmch 29012 hmopidmpj 29013 pjss1coi 29022 padct 29497 locfinreflem 29907 locfinref 29908 issibf 30395 sitgfval 30403 mrsubvrs 31419 rdgprc0 31699 rdgprc 31700 dfrdg2 31701 madeval 31935 brrangeg 32043 poimirlem24 33433 volsupnfl 33454 elghomlem1OLD 33684 isdivrngo 33749 iscom2 33794 dnnumch1 37614 aomclem3 37626 aomclem8 37631 rclexi 37922 rtrclex 37924 rtrclexi 37928 cnvrcl0 37932 dfrtrcl5 37936 dfrcl2 37966 csbima12gALTVD 39133 unirnmap 39400 ssmapsn 39408 sge0val 40583 vonvolmbl 40875 |
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