Theorem op1st2nd 5791

Description: Express equality to an ordered pair via 1^st and 2^nd. (Contributed by SF, 12-Feb-2015.)

Hypotheses

Ref	Expression
op1st2nd.1	⊢ A ∈ V
op1st2nd.2	⊢ B ∈ V

Assertion

Ref	Expression
op1st2nd	⊢ ((C1st A ∧ C2nd B) ↔ C = ⟨A, B⟩)

Proof of Theorem op1st2nd

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		op1st2nd.1	. . . . 5 ⊢ A ∈ V
2	1	br1st 4859	. . . 4 ⊢ (C1st A ↔ ∃x C = ⟨A, x⟩)
3		vex 2863	. . . . . . . . 9 ⊢ x ∈ V
4	1, 3	opbr2nd 5503	. . . . . . . 8 ⊢ (⟨A, x⟩2nd B ↔ x = B)
5	4	biimpi 186	. . . . . . 7 ⊢ (⟨A, x⟩2nd B → x = B)
6	5	opeq2d 4586	. . . . . 6 ⊢ (⟨A, x⟩2nd B → ⟨A, x⟩ = ⟨A, B⟩)
7		breq1 4643	. . . . . . 7 ⊢ (C = ⟨A, x⟩ → (C2nd B ↔ ⟨A, x⟩2nd B))
8		eqeq1 2359	. . . . . . 7 ⊢ (C = ⟨A, x⟩ → (C = ⟨A, B⟩ ↔ ⟨A, x⟩ = ⟨A, B⟩))
9	7, 8	imbi12d 311	. . . . . 6 ⊢ (C = ⟨A, x⟩ → ((C2nd B → C = ⟨A, B⟩) ↔ (⟨A, x⟩2nd B → ⟨A, x⟩ = ⟨A, B⟩)))
10	6, 9	mpbiri 224	. . . . 5 ⊢ (C = ⟨A, x⟩ → (C2nd B → C = ⟨A, B⟩))
11	10	exlimiv 1634	. . . 4 ⊢ (∃x C = ⟨A, x⟩ → (C2nd B → C = ⟨A, B⟩))
12	2, 11	sylbi 187	. . 3 ⊢ (C1st A → (C2nd B → C = ⟨A, B⟩))
13	12	imp 418	. 2 ⊢ ((C1st A ∧ C2nd B) → C = ⟨A, B⟩)
14		eqid 2353	. . . . 5 ⊢ A = A
15		op1st2nd.2	. . . . . 6 ⊢ B ∈ V
16	1, 15	opbr1st 5502	. . . . 5 ⊢ (⟨A, B⟩1st A ↔ A = A)
17	14, 16	mpbir 200	. . . 4 ⊢ ⟨A, B⟩1st A
18		eqid 2353	. . . . 5 ⊢ B = B
19	1, 15	opbr2nd 5503	. . . . 5 ⊢ (⟨A, B⟩2nd B ↔ B = B)
20	18, 19	mpbir 200	. . . 4 ⊢ ⟨A, B⟩2nd B
21	17, 20	pm3.2i 441	. . 3 ⊢ (⟨A, B⟩1st A ∧ ⟨A, B⟩2nd B)
22		breq1 4643	. . . 4 ⊢ (C = ⟨A, B⟩ → (C1st A ↔ ⟨A, B⟩1st A))
23		breq1 4643	. . . 4 ⊢ (C = ⟨A, B⟩ → (C2nd B ↔ ⟨A, B⟩2nd B))
24	22, 23	anbi12d 691	. . 3 ⊢ (C = ⟨A, B⟩ → ((C1st A ∧ C2nd B) ↔ (⟨A, B⟩1st A ∧ ⟨A, B⟩2nd B)))
25	21, 24	mpbiri 224	. 2 ⊢ (C = ⟨A, B⟩ → (C1st A ∧ C2nd B))
26	13, 25	impbii 180	1 ⊢ ((C1st A ∧ C2nd B) ↔ C = ⟨A, B⟩)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860  ⟨cop 4562   class class class wbr 4640  1^st c1st 4718  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-2nd 4798

This theorem is referenced by:  otsnelsi3  5806  composeex  5821  addcfnex  5825  funsex  5829  crossex  5851  transex  5911  refex  5912  antisymex  5913  enpw1lem1  6062  enmap2lem1  6064  enmap1lem1  6070  enprmaplem1  6077  ovmuc  6131  ce0nn  6181  nncdiv3lem1  6276  nncdiv3lem2  6277  nnc3n3p1  6279  spacvallem1  6282  nchoicelem16  6305