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Mirrors > Home > MPE Home > Th. List > feq23d | Structured version Visualization version GIF version |
Description: Equality deduction for functions. (Contributed by NM, 8-Jun-2013.) |
Ref | Expression |
---|---|
feq23d.1 | ⊢ (𝜑 → 𝐴 = 𝐶) |
feq23d.2 | ⊢ (𝜑 → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
feq23d | ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2736 | . 2 ⊢ (𝜑 → 𝐹 = 𝐹) | |
2 | feq23d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐶) | |
3 | feq23d.2 | . 2 ⊢ (𝜑 → 𝐵 = 𝐷) | |
4 | 1, 2, 3 | feq123d 6726 | 1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ⟶wf 6559 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-fun 6565 df-fn 6566 df-f 6567 |
This theorem is referenced by: nvof1o 7300 axdc4uz 14022 isacs 17696 isfunc 17915 funcres 17947 funcpropd 17954 estrcco 18185 funcestrcsetclem9 18204 fullestrcsetc 18207 fullsetcestrc 18222 1stfcl 18253 2ndfcl 18254 evlfcl 18279 curf1cl 18285 yonedalem3b 18336 intopsn 18680 mgmhmpropd 18724 mhmpropd 18818 isghm 19246 pwssplit1 21076 islindf 21850 evls1sca 22343 rrxds 25441 wlkp1 29714 acunirnmpt 32676 fnpreimac 32688 pwrssmgc 32975 cnmbfm 34245 elmrsubrn 35505 poimirlem3 37610 poimirlem28 37635 isrngod 37885 rngosn3 37911 isgrpda 37942 islfld 39044 tendofset 40741 tendoset 40742 sn-isghm 42660 mapfzcons 42704 diophrw 42747 refsum2cnlem1 44975 funcringcsetcALTV2lem9 48142 funcringcsetclem9ALTV 48165 aacllem 49032 |
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