Theorem xp2nd 6334

Description: Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
xp2nd	⊢ (𝐴 ∈ (𝐵 × 𝐶) → (2nd ‘𝐴) ∈ 𝐶)

Proof of Theorem xp2nd

Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp 4744	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑏∃𝑐(𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)))
2		vex 2804	. . . . . . 7 ⊢ 𝑏 ∈ V
3		vex 2804	. . . . . . 7 ⊢ 𝑐 ∈ V
4	2, 3	op2ndd 6317	. . . . . 6 ⊢ (𝐴 = ⟨𝑏, 𝑐⟩ → (2nd ‘𝐴) = 𝑐)
5	4	eleq1d 2299	. . . . 5 ⊢ (𝐴 = ⟨𝑏, 𝑐⟩ → ((2nd ‘𝐴) ∈ 𝐶 ↔ 𝑐 ∈ 𝐶))
6	5	biimpar 297	. . . 4 ⊢ ((𝐴 = ⟨𝑏, 𝑐⟩ ∧ 𝑐 ∈ 𝐶) → (2nd ‘𝐴) ∈ 𝐶)
7	6	adantrl 478	. . 3 ⊢ ((𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (2nd ‘𝐴) ∈ 𝐶)
8	7	exlimivv 1944	. 2 ⊢ (∃𝑏∃𝑐(𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (2nd ‘𝐴) ∈ 𝐶)
9	1, 8	sylbi 121	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (2nd ‘𝐴) ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397  ∃wex 1540   ∈ wcel 2201  ⟨cop 3673   × cxp 4725  ‘cfv 5328  2^nd c2nd 6307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-sbc 3031  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-iota 5288  df-fun 5330  df-fv 5336  df-2nd 6309

This theorem is referenced by:  xpf1o  7035  xpmapenlem  7040  opabfi  7137  djuf1olem  7257  exmidapne  7484  cc2lem  7490  dfplpq2  7579  dfmpq2  7580  enqbreq2  7582  enqdc1  7587  mulpipq2  7596  preqlu  7697  elnp1st2nd  7701  cauappcvgprlemladd  7883  elreal2  8055  cnref1o  9890  frecuzrdgrrn  10676  frec2uzrdg  10677  frecuzrdgrcl  10678  frecuzrdgtcl  10680  frecuzrdgsuc  10682  frecuzrdgrclt  10683  frecuzrdgg  10684  frecuzrdgdomlem  10685  frecuzrdgfunlem  10687  frecuzrdgsuctlem  10691  seq3val  10728  seqvalcd  10729  fisumcom2  12022  fprodcom2fi  12210  eucalgval  12649  eucalginv  12651  eucalglt  12652  eucalgcvga  12653  eucalg  12654  sqpweven  12770  2sqpwodd  12771  ctiunctlemudc  13081  xpsff1o  13455  tx1cn  15022  txdis  15030  txhmeo  15072  xmetxp  15260  xmetxpbl  15261  xmettxlem  15262  xmettx  15263