| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 2738 | . 2 ⊢ (𝜑 → 𝐹 = 𝐹) | |
| 2 | feq23d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐶) | |
| 3 | feq23d.2 | . 2 ⊢ (𝜑 → 𝐵 = 𝐷) | |
| 4 | 1, 2, 3 | feq123d 6725 | 1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ⟶wf 6557 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 |
| This theorem is referenced by: nvof1o 7300 axdc4uz 14025 isacs 17694 isfunc 17909 funcres 17941 funcpropd 17947 estrcco 18174 funcestrcsetclem9 18193 fullestrcsetc 18196 fullsetcestrc 18211 1stfcl 18242 2ndfcl 18243 evlfcl 18267 curf1cl 18273 yonedalem3b 18324 intopsn 18667 mgmhmpropd 18711 mhmpropd 18805 isghm 19233 pwssplit1 21058 islindf 21832 evls1sca 22327 rrxds 25427 wlkp1 29699 acunirnmpt 32669 fnpreimac 32681 pwrssmgc 32990 cnmbfm 34265 elmrsubrn 35525 poimirlem3 37630 poimirlem28 37655 isrngod 37905 rngosn3 37931 isgrpda 37962 islfld 39063 tendofset 40760 tendoset 40761 sn-isghm 42683 mapfzcons 42727 diophrw 42770 refsum2cnlem1 45042 funcringcsetcALTV2lem9 48214 funcringcsetclem9ALTV 48237 aacllem 49320 |
| Copyright terms: Public domain | W3C validator |