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Mirrors > Home > MPE Home > Th. List > fneq1d | Structured version Visualization version GIF version |
Description: Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
fneq1d.1 | ⊢ (𝜑 → 𝐹 = 𝐺) |
Ref | Expression |
---|---|
fneq1d | ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fneq1d.1 | . 2 ⊢ (𝜑 → 𝐹 = 𝐺) | |
2 | fneq1 5893 | . 2 ⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 Fn wfn 5799 |
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-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-fun 5806 df-fn 5807 |
This theorem is referenced by: fneq12d 5897 f1o00 6083 f1oprswap 6092 f1ompt 6290 fmpt2d 6300 f1ocnvd 6782 offn 6806 offval2f 6807 offval2 6812 ofrfval2 6813 caofinvl 6822 suppsnop 7196 omxpenlem 7946 itunifn 9122 konigthlem 9269 seqof 12720 swrdlen 13275 mptfzshft 14352 fsumrev 14353 fprodrev 14546 prdsdsfn 15948 imasdsfn 15997 xpscfn 16042 cidfn 16163 comffn 16188 isoval 16248 invf1o 16252 isofn 16258 brssc 16297 cofucl 16371 estrchomfn 16598 funcestrcsetclem4 16606 funcsetcestrclem4 16621 1stfcl 16660 2ndfcl 16661 prfcl 16666 evlfcl 16685 curf1cl 16691 curfcl 16695 hofcl 16722 yonedainv 16744 grpinvf1o 17308 pmtrrn 17700 pmtrfrn 17701 srngf1o 18677 ofco2 20076 mat1dimscm 20100 neif 20714 fmf 21559 fncpn 23502 mdeg0 23634 tglnfn 25242 grpoinvf 26770 kbass2 28360 fnresin 28812 f1o3d 28813 f1od2 28887 pstmxmet 29268 ofcfn 29489 ofcfval2 29493 signstlen 29970 bnj941 30097 msubrn 30680 poimirlem4 32583 cnambfre 32628 sdclem2 32708 diafn 35341 dibfna 35461 dicfnN 35490 dihf11lem 35573 mapd1o 35955 hdmapfnN 36139 hgmapfnN 36198 hbtlem7 36714 fsovf1od 37330 ntrrn 37440 ntrf 37441 dssmapntrcls 37446 addrfn 37697 subrfn 37698 mulvfn 37699 fsumsermpt 38646 hoidmvlelem3 39487 rnghmresfn 41755 rhmresfn 41801 funcringcsetcALTV2lem4 41831 funcringcsetclem4ALTV 41854 rhmsubclem1 41878 rhmsubcALTVlem1 41897 |
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