Theorem 1st2nd2 5517

Description: Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by SF, 20-Oct-2013.)

Assertion

Ref	Expression
1st2nd2	⊢ (A ∈ (B × C) → A = ⟨(1st ‘A), (2nd ‘A)⟩)

Proof of Theorem 1st2nd2

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

Step	Hyp	Ref	Expression
1		elxp2 4803	. 2 ⊢ (A ∈ (B × C) ↔ ∃x ∈ B ∃y ∈ C A = ⟨x, y⟩)
2		vex 2863	. . . . . . . 8 ⊢ x ∈ V
3		vex 2863	. . . . . . . 8 ⊢ y ∈ V
4	2, 3	opfv1st 5515	. . . . . . 7 ⊢ (1st ‘⟨x, y⟩) = x
5	2, 3	opfv2nd 5516	. . . . . . 7 ⊢ (2nd ‘⟨x, y⟩) = y
6	4, 5	opeq12i 4584	. . . . . 6 ⊢ ⟨(1st ‘⟨x, y⟩), (2nd ‘⟨x, y⟩)⟩ = ⟨x, y⟩
7	6	eqcomi 2357	. . . . 5 ⊢ ⟨x, y⟩ = ⟨(1st ‘⟨x, y⟩), (2nd ‘⟨x, y⟩)⟩
8		id 19	. . . . 5 ⊢ (A = ⟨x, y⟩ → A = ⟨x, y⟩)
9		fveq2 5329	. . . . . 6 ⊢ (A = ⟨x, y⟩ → (1st ‘A) = (1st ‘⟨x, y⟩))
10		fveq2 5329	. . . . . 6 ⊢ (A = ⟨x, y⟩ → (2nd ‘A) = (2nd ‘⟨x, y⟩))
11	9, 10	opeq12d 4587	. . . . 5 ⊢ (A = ⟨x, y⟩ → ⟨(1st ‘A), (2nd ‘A)⟩ = ⟨(1st ‘⟨x, y⟩), (2nd ‘⟨x, y⟩)⟩)
12	7, 8, 11	3eqtr4a 2411	. . . 4 ⊢ (A = ⟨x, y⟩ → A = ⟨(1st ‘A), (2nd ‘A)⟩)
13	12	rexlimivw 2735	. . 3 ⊢ (∃y ∈ C A = ⟨x, y⟩ → A = ⟨(1st ‘A), (2nd ‘A)⟩)
14	13	rexlimivw 2735	. 2 ⊢ (∃x ∈ B ∃y ∈ C A = ⟨x, y⟩ → A = ⟨(1st ‘A), (2nd ‘A)⟩)
15	1, 14	sylbi 187	1 ⊢ (A ∈ (B × C) → A = ⟨(1st ‘A), (2nd ‘A)⟩)

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  ⟨cop 4562  1^st c1st 4718   × cxp 4771   ‘cfv 4782  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-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-fo 4794  df-fv 4796  df-2nd 4798

This theorem is referenced by: (None)