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Mirrors > Home > MPE Home > Th. List > prf2 | Structured version Visualization version Unicode version |
Description: Value of the pairing functor on morphisms. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Ref | Expression |
---|---|
prfval.k | 〈,〉F |
prfval.b | |
prfval.h | |
prfval.c | |
prfval.d | |
prf1.x | |
prf2.y | |
prf2.k |
Ref | Expression |
---|---|
prf2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prfval.k | . . 3 〈,〉F | |
2 | prfval.b | . . 3 | |
3 | prfval.h | . . 3 | |
4 | prfval.c | . . 3 | |
5 | prfval.d | . . 3 | |
6 | prf1.x | . . 3 | |
7 | prf2.y | . . 3 | |
8 | 1, 2, 3, 4, 5, 6, 7 | prf2fval 16841 | . 2 |
9 | simpr 477 | . . . 4 | |
10 | 9 | fveq2d 6195 | . . 3 |
11 | 9 | fveq2d 6195 | . . 3 |
12 | 10, 11 | opeq12d 4410 | . 2 |
13 | prf2.k | . 2 | |
14 | opex 4932 | . . 3 | |
15 | 14 | a1i 11 | . 2 |
16 | 8, 12, 13, 15 | fvmptd 6288 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cvv 3200 cop 4183 cfv 5888 (class class class)co 6650 c2nd 7167 cbs 15857 chom 15952 cfunc 16514 〈,〉F cprf 16811 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 df-ixp 7909 df-func 16518 df-prf 16815 |
This theorem is referenced by: prfcl 16843 prf1st 16844 prf2nd 16845 uncf2 16877 yonedalem22 16918 |
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