Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > feq12d | Structured version Visualization version GIF version |
Description: Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
feq12d.1 | ⊢ (𝜑 → 𝐹 = 𝐺) |
feq12d.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
feq12d | ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq12d.1 | . . 3 ⊢ (𝜑 → 𝐹 = 𝐺) | |
2 | 1 | feq1d 6585 | . 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐴⟶𝐶)) |
3 | feq12d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | feq2d 6586 | . 2 ⊢ (𝜑 → (𝐺:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶)) |
5 | 2, 4 | bitrd 278 | 1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ⟶wf 6429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-fun 6435 df-fn 6436 df-f 6437 |
This theorem is referenced by: feq123d 6589 fprg 7027 smoeq 8181 oif 9289 1fv 13375 catcisolem 17825 hofcl 17977 dmdprd 19601 dpjf 19660 pjf2 20921 mat1dimmul 21625 lmbr2 22410 lmff 22452 dfac14 22769 lmmbr2 24423 lmcau 24477 perfdvf 25067 dvnfre 25116 dvle 25171 dvfsumle 25185 dvfsumge 25186 dvmptrecl 25188 uhgr0e 27441 uhgrstrrepe 27448 incistruhgr 27449 upgr1e 27483 1hevtxdg1 27873 umgr2v2e 27892 iswlk 27977 0wlkons1 28485 resf1o 31065 ismeas 32167 omsmeas 32290 breprexplema 32610 satfun 33373 mbfresfi 35823 sdclem1 35901 dfac21 40891 fnlimfvre 43215 climrescn 43289 fourierdlem74 43721 fourierdlem103 43750 fourierdlem104 43751 sge0iunmpt 43956 ismea 43989 isome 44032 sssmf 44274 smflimlem3 44308 smflimlem4 44309 isupwlk 45298 |
Copyright terms: Public domain | W3C validator |