Theorem df2nd2 5112

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

Assertion

Ref	Expression
df2nd2	⊢ 2nd = (1st ∘ 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 ∈ V
2	1	br1st 4859	. . . . . . 7 ⊢ (w1st y ↔ ∃z w = ⟨y, z⟩)
3	2	anbi1i 676	. . . . . 6 ⊢ ((w1st y ∧ x Swap w) ↔ (∃z w = ⟨y, z⟩ ∧ x Swap w))
4		ancom 437	. . . . . 6 ⊢ ((x Swap w ∧ w1st y) ↔ (w1st y ∧ x Swap w))
5		19.41v 1901	. . . . . 6 ⊢ (∃z(w = ⟨y, z⟩ ∧ x Swap w) ↔ (∃z w = ⟨y, z⟩ ∧ x Swap w))
6	3, 4, 5	3bitr4i 268	. . . . 5 ⊢ ((x Swap w ∧ w1st y) ↔ ∃z(w = ⟨y, z⟩ ∧ x Swap w))
7	6	exbii 1582	. . . 4 ⊢ (∃w(x Swap w ∧ w1st y) ↔ ∃w∃z(w = ⟨y, z⟩ ∧ x Swap w))
8		excom 1741	. . . 4 ⊢ (∃z∃w(w = ⟨y, z⟩ ∧ x Swap w) ↔ ∃w∃z(w = ⟨y, z⟩ ∧ x Swap w))
9		vex 2863	. . . . . . . 8 ⊢ z ∈ V
10	1, 9	opex 4589	. . . . . . 7 ⊢ ⟨y, z⟩ ∈ V
11		breq2 4644	. . . . . . 7 ⊢ (w = ⟨y, z⟩ → (x Swap w ↔ x Swap ⟨y, z⟩))
12	10, 11	ceqsexv 2895	. . . . . 6 ⊢ (∃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 ⊢ (∃w(w = ⟨y, z⟩ ∧ x Swap w) ↔ x = ⟨z, y⟩)
15	14	exbii 1582	. . . 4 ⊢ (∃z∃w(w = ⟨y, z⟩ ∧ x Swap w) ↔ ∃z x = ⟨z, y⟩)
16	7, 8, 15	3bitr2ri 265	. . 3 ⊢ (∃z x = ⟨z, y⟩ ↔ ∃w(x Swap w ∧ w1st y))
17	16	opabbii 4627	. 2 ⊢ {⟨x, y⟩ ∣ ∃z x = ⟨z, y⟩} = {⟨x, y⟩ ∣ ∃w(x Swap w ∧ w1st y)}
18		df-2nd 4798	. 2 ⊢ 2nd = {⟨x, y⟩ ∣ ∃z x = ⟨z, y⟩}
19		df-co 4727	. 2 ⊢ (1st ∘ Swap ) = {⟨x, y⟩ ∣ ∃w(x Swap w ∧ w1st y)}
20	17, 18, 19	3eqtr4i 2383	1 ⊢ 2nd = (1st ∘ Swap )

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642  ⟨cop 4562  {copab 4623   class class class wbr 4640  1^st c1st 4718   Swap cswap 4719   ∘ ccom 4722  2^nd c2nd 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