Theorem df2nd2 5111

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 2862	. . . . . . . 8 ⊢ y ∈ V
2	1	br1st 4858	. . . . . . 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 2862	. . . . . . . 8 ⊢ z ∈ V
10	1, 9	opex 4588	. . . . . . 7 ⊢ ⟨y, z⟩ ∈ V
11		breq2 4643	. . . . . . 7 ⊢ (w = ⟨y, z⟩ → (x Swap w ↔ x Swap ⟨y, z⟩))
12	10, 11	ceqsexv 2894	. . . . . 6 ⊢ (∃w(w = ⟨y, z⟩ ∧ x Swap w) ↔ x Swap ⟨y, z⟩)
13	1, 9	brswap2 4860	. . . . . 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 4626	. 2 ⊢ {⟨x, y⟩ ∣ ∃z x = ⟨z, y⟩} = {⟨x, y⟩ ∣ ∃w(x Swap w ∧ w1st y)}
18		df-2nd 4797	. 2 ⊢ 2nd = {⟨x, y⟩ ∣ ∃z x = ⟨z, y⟩}
19		df-co 4726	. 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 4561  {copab 4622   class class class wbr 4639  1^st c1st 4717   Swap cswap 4718   ∘ ccom 4721  2^nd c2nd 4783

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-co 4726  df-2nd 4797

This theorem is referenced by:  2ndex  5112