Theorem xp1st 6258

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

Assertion

Ref	Expression
xp1st	⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1st ‘𝐴) ∈ 𝐵)

Proof of Theorem xp1st

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

Step	Hyp	Ref	Expression
1		elxp 4696	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑏∃𝑐(𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)))
2		vex 2776	. . . . . . 7 ⊢ 𝑏 ∈ V
3		vex 2776	. . . . . . 7 ⊢ 𝑐 ∈ V
4	2, 3	op1std 6241	. . . . . 6 ⊢ (𝐴 = ⟨𝑏, 𝑐⟩ → (1st ‘𝐴) = 𝑏)
5	4	eleq1d 2275	. . . . 5 ⊢ (𝐴 = ⟨𝑏, 𝑐⟩ → ((1st ‘𝐴) ∈ 𝐵 ↔ 𝑏 ∈ 𝐵))
6	5	biimpar 297	. . . 4 ⊢ ((𝐴 = ⟨𝑏, 𝑐⟩ ∧ 𝑏 ∈ 𝐵) → (1st ‘𝐴) ∈ 𝐵)
7	6	adantrr 479	. . 3 ⊢ ((𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (1st ‘𝐴) ∈ 𝐵)
8	7	exlimivv 1921	. 2 ⊢ (∃𝑏∃𝑐(𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (1st ‘𝐴) ∈ 𝐵)
9	1, 8	sylbi 121	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1st ‘𝐴) ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373  ∃wex 1516   ∈ wcel 2177  ⟨cop 3637   × cxp 4677  ‘cfv 5276  1^st c1st 6231

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3000  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-iota 5237  df-fun 5278  df-fv 5284  df-1st 6233

This theorem is referenced by:  disjxp1  6329  xpf1o  6948  xpmapenlem  6953  opabfi  7042  djuf1olem  7162  eldju1st  7180  exmidapne  7379  dfplpq2  7474  dfmpq2  7475  enqbreq2  7477  enqdc1  7482  mulpipq2  7491  preqlu  7592  elnp1st2nd  7596  cauappcvgprlemladd  7778  elreal2  7950  cnref1o  9779  frecuzrdgrrn  10560  frec2uzrdg  10561  frecuzrdgrcl  10562  frecuzrdgsuc  10566  frecuzrdgrclt  10567  frecuzrdgg  10568  frecuzrdgsuctlem  10575  seq3val  10612  seqvalcd  10613  fsum2dlemstep  11789  fisumcom2  11793  fprod2dlemstep  11977  fprodcom2fi  11981  eucalgval  12420  eucalginv  12422  eucalglt  12423  eucalg  12425  sqpweven  12541  2sqpwodd  12542  ctiunctlemudc  12852  xpsff1o  13225  tx2cn  14786  txdis  14793  txhmeo  14835  xmetxp  15023  xmetxpbl  15024  xmettxlem  15025  xmettx  15026  lgsquadlemofi  15597  lgsquadlem1  15598  lgsquadlem2  15599