![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > funeqd | Structured version Visualization version GIF version |
Description: Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.) |
Ref | Expression |
---|---|
funeqd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
funeqd | ⊢ (𝜑 → (Fun 𝐴 ↔ Fun 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funeqd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | funeq 6587 | . 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (Fun 𝐴 ↔ Fun 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1536 Fun wfun 6556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ss 3979 df-br 5148 df-opab 5210 df-rel 5695 df-cnv 5696 df-co 5697 df-fun 6564 |
This theorem is referenced by: funopg 6601 funsng 6618 f1eq1 6799 f1ssf1 6880 fvn0ssdmfun 7093 funcnvuni 7954 fundmge2nop0 14537 funcnvs2 14948 funcnvs3 14949 funcnvs4 14950 shftfn 15108 isstruct2 17182 structfung 17187 strle1 17191 setsfun 17204 setsfun0 17205 monfval 17779 ismon 17780 monpropd 17784 isepi 17787 isfth 17967 estrres 18194 lubfun 18409 glbfun 18422 acsficl2d 18609 ebtwntg 29011 ecgrtg 29012 elntg 29013 uhgrspansubgrlem 29321 istrl 29728 ispth 29755 isspth 29756 upgrwlkdvspth 29771 uhgrwkspthlem1 29785 uhgrwkspthlem2 29786 usgr2wlkspthlem1 29789 usgr2wlkspthlem2 29790 pthdlem1 29798 2spthd 29970 0spth 30154 3spthd 30204 trlsegvdeglem2 30249 trlsegvdeglem3 30250 ajfun 30888 fresf1o 32647 padct 32736 smatrcl 33756 esum2dlem 34072 omssubadd 34281 sitgf 34328 funen1cnv 35080 pthhashvtx 35111 satfv0fun 35355 satffunlem1 35391 satffunlem2 35392 satffun 35393 satefvfmla0 35402 satefvfmla1 35409 fperdvper 45874 ovnovollem1 46611 funressnmo 46995 dfateq12d 47075 afvres 47121 funressndmafv2rn 47172 afv2res 47188 fdivval 48388 |
Copyright terms: Public domain | W3C validator |