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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 2728 | . 2 ⊢ (𝜑 → 𝐹 = 𝐹) | |
2 | feq23d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐶) | |
3 | feq23d.2 | . 2 ⊢ (𝜑 → 𝐵 = 𝐷) | |
4 | 1, 2, 3 | feq123d 6705 | 1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ⟶wf 6538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-fun 6544 df-fn 6545 df-f 6546 |
This theorem is referenced by: nvof1o 7283 axdc4uz 13973 isacs 17622 isfunc 17841 funcres 17873 funcpropd 17880 estrcco 18111 funcestrcsetclem9 18130 fullestrcsetc 18133 fullsetcestrc 18148 1stfcl 18179 2ndfcl 18180 evlfcl 18205 curf1cl 18211 yonedalem3b 18262 intopsn 18605 mgmhmpropd 18649 mhmpropd 18740 pwssplit1 20933 islindf 21733 evls1sca 22229 rrxds 25308 wlkp1 29482 acunirnmpt 32428 fnpreimac 32440 pwrssmgc 32709 cnmbfm 33819 elmrsubrn 35066 poimirlem3 37031 poimirlem28 37056 isrngod 37306 rngosn3 37332 isgrpda 37363 islfld 38471 tendofset 40168 tendoset 40169 mapfzcons 42058 diophrw 42101 refsum2cnlem1 44322 funcringcsetcALTV2lem9 47283 funcringcsetclem9ALTV 47306 aacllem 48157 |
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