Theorem df2nd2 5112

Description: Alternate definition of the 2nd function. (Contributed by SF, 8-Jan-2015.)

Assertion

Ref	Expression
df2nd2	|- 2nd = (1st o. Swap )

Proof of Theorem df2nd2

Dummy variables x y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2863	. . . . . . . 8 |- y e. _V
2	1	br1st 4859	. . . . . . 7 |- (w1sty <-> E.z w = <.y, z>.)
3	2	anbi1i 676	. . . . . 6 |- ((w1sty /\ x Swap w) <-> (E.z w = <.y, z>. /\ x Swap w))
4		ancom 437	. . . . . 6 |- ((x Swap w /\ w1sty) <-> (w1sty /\ x Swap w))
5		19.41v 1901	. . . . . 6 |- (E.z(w = <.y, z>. /\ x Swap w) <-> (E.z w = <.y, z>. /\ x Swap w))
6	3, 4, 5	3bitr4i 268	. . . . 5 |- ((x Swap w /\ w1sty) <-> E.z(w = <.y, z>. /\ x Swap w))
7	6	exbii 1582	. . . 4 |- (E.w(x Swap w /\ w1sty) <-> E.wE.z(w = <.y, z>. /\ x Swap w))
8		excom 1741	. . . 4 |- (E.zE.w(w = <.y, z>. /\ x Swap w) <-> E.wE.z(w = <.y, z>. /\ x Swap w))
9		vex 2863	. . . . . . . 8 |- z e. _V
10	1, 9	opex 4589	. . . . . . 7 |- <.y, z>. e. _V
11		breq2 4644	. . . . . . 7 |- (w = <.y, z>. -> (x Swap w <-> x Swap <.y, z>.))
12	10, 11	ceqsexv 2895	. . . . . 6 |- (E.w(w = <.y, z>. /\ x Swap w) <-> x Swap <.y, z>.)
13	1, 9	brswap2 4861	. . . . . 6 |- (x Swap <.y, z>. <-> x = <.z, y>.)
14	12, 13	bitri 240	. . . . 5 |- (E.w(w = <.y, z>. /\ x Swap w) <-> x = <.z, y>.)
15	14	exbii 1582	. . . 4 |- (E.zE.w(w = <.y, z>. /\ x Swap w) <-> E.z x = <.z, y>.)
16	7, 8, 15	3bitr2ri 265	. . 3 |- (E.z x = <.z, y>. <-> E.w(x Swap w /\ w1sty))
17	16	opabbii 4627	. 2 |- {<.x, y>. | E.z x = <.z, y>.} = {<.x, y>. | E.w(x Swap w /\ w1sty)}
18		df-2nd 4798	. 2 |- 2nd = {<.x, y>. | E.z x = <.z, y>.}
19		df-co 4727	. 2 |- (1st o. Swap ) = {<.x, y>. | E.w(x Swap w /\ w1sty)}
20	17, 18, 19	3eqtr4i 2383	1 |- 2nd = (1st o. Swap )

Syntax hints:   /\ wa 358  E.wex 1541   = wceq 1642  <.cop 4562  {copab 4623   class class class wbr 4640  1stc1st 4718   Swap cswap 4719   o. ccom 4722  2ndc2nd 4784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-co 4727  df-2nd 4798

This theorem is referenced by:  2ndex  5113