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Mirrors > Home > MPE Home > Th. List > feq1 | Structured version Visualization version GIF version |
Description: Equality theorem for functions. (Contributed by NM, 1-Aug-1994.) |
Ref | Expression |
---|---|
feq1 | ⊢ (𝐹 = 𝐺 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fneq1 6439 | . . 3 ⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) | |
2 | rneq 5801 | . . . 4 ⊢ (𝐹 = 𝐺 → ran 𝐹 = ran 𝐺) | |
3 | 2 | sseq1d 3998 | . . 3 ⊢ (𝐹 = 𝐺 → (ran 𝐹 ⊆ 𝐵 ↔ ran 𝐺 ⊆ 𝐵)) |
4 | 1, 3 | anbi12d 632 | . 2 ⊢ (𝐹 = 𝐺 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵))) |
5 | df-f 6354 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
6 | df-f 6354 | . 2 ⊢ (𝐺:𝐴⟶𝐵 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵)) | |
7 | 4, 5, 6 | 3bitr4g 316 | 1 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ⊆ wss 3936 ran crn 5551 Fn wfn 6345 ⟶wf 6346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-fun 6352 df-fn 6353 df-f 6354 |
This theorem is referenced by: feq1d 6494 feq1i 6500 elimf 6508 f00 6556 f0bi 6557 f0dom0 6558 fconstg 6561 f1eq1 6565 fprb 6951 fconst2g 6960 fsnex 7033 elmapg 8413 ac6sfi 8756 updjud 9357 ac5num 9456 acni2 9466 cofsmo 9685 cfsmolem 9686 cfcoflem 9688 coftr 9689 alephsing 9692 axdc2lem 9864 axdc3lem2 9867 axdc3lem3 9868 axdc3 9870 axdc4lem 9871 ac6num 9895 inar1 10191 axdc4uzlem 13345 seqf1olem2 13404 seqf1o 13405 iswrd 13857 cshf1 14166 wrdlen2i 14298 ramub2 16344 ramcl 16359 isacs2 16918 isacs1i 16922 mreacs 16923 mgmb1mgm1 17859 elefmndbas2 18033 isgrpinv 18150 isghm 18352 islindf 20950 mat1dimelbas 21074 1stcfb 22047 upxp 22225 txcn 22228 isi1f 24269 mbfi1fseqlem6 24315 mbfi1flimlem 24317 itg2addlem 24353 plyf 24782 griedg0prc 27040 isgrpo 28268 vciOLD 28332 isvclem 28348 isnvlem 28381 ajmoi 28629 ajval 28632 hlimi 28959 chlimi 29005 chcompl 29013 adjmo 29603 adjeu 29660 adjval 29661 adj1 29704 adjeq 29706 cnlnssadj 29851 pjinvari 29962 padct 30449 locfinref 31100 isrnmeas 31454 orderseqlem 33089 soseq 33091 elno 33148 filnetlem4 33724 bj-finsumval0 34561 poimirlem25 34911 poimirlem28 34914 volsupnfl 34931 mbfresfi 34932 upixp 34998 sdclem2 35011 sdclem1 35012 fdc 35014 ismgmOLD 35122 elghomlem2OLD 35158 istendo 37890 ismrc 39291 fmuldfeqlem1 41855 fmuldfeq 41856 dvnprodlem1 42223 stoweidlem15 42293 stoweidlem16 42294 stoweidlem17 42295 stoweidlem19 42297 stoweidlem20 42298 stoweidlem21 42299 stoweidlem22 42300 stoweidlem23 42301 stoweidlem27 42305 stoweidlem31 42309 stoweidlem32 42310 stoweidlem42 42320 stoweidlem48 42326 stoweidlem51 42329 stoweidlem59 42337 isomenndlem 42805 smfpimcclem 43074 lincdifsn 44472 |
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