Theorem xp1st 6144

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 4628	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑏∃𝑐(𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)))
2		vex 2733	. . . . . . 7 ⊢ 𝑏 ∈ V
3		vex 2733	. . . . . . 7 ⊢ 𝑐 ∈ V
4	2, 3	op1std 6127	. . . . . 6 ⊢ (𝐴 = ⟨𝑏, 𝑐⟩ → (1st ‘𝐴) = 𝑏)
5	4	eleq1d 2239	. . . . 5 ⊢ (𝐴 = ⟨𝑏, 𝑐⟩ → ((1st ‘𝐴) ∈ 𝐵 ↔ 𝑏 ∈ 𝐵))
6	5	biimpar 295	. . . 4 ⊢ ((𝐴 = ⟨𝑏, 𝑐⟩ ∧ 𝑏 ∈ 𝐵) → (1st ‘𝐴) ∈ 𝐵)
7	6	adantrr 476	. . 3 ⊢ ((𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (1st ‘𝐴) ∈ 𝐵)
8	7	exlimivv 1889	. 2 ⊢ (∃𝑏∃𝑐(𝐴 = ⟨𝑏, 𝑐⟩ ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (1st ‘𝐴) ∈ 𝐵)
9	1, 8	sylbi 120	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1st ‘𝐴) ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348  ∃wex 1485   ∈ wcel 2141  ⟨cop 3586   × cxp 4609  ‘cfv 5198  1^st c1st 6117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fv 5206  df-1st 6119

This theorem is referenced by:  disjxp1  6215  xpf1o  6822  xpmapenlem  6827  djuf1olem  7030  eldju1st  7048  dfplpq2  7316  dfmpq2  7317  enqbreq2  7319  enqdc1  7324  mulpipq2  7333  preqlu  7434  elnp1st2nd  7438  cauappcvgprlemladd  7620  elreal2  7792  cnref1o  9609  frecuzrdgrrn  10364  frec2uzrdg  10365  frecuzrdgrcl  10366  frecuzrdgsuc  10370  frecuzrdgrclt  10371  frecuzrdgg  10372  frecuzrdgsuctlem  10379  seq3val  10414  seqvalcd  10415  fsum2dlemstep  11397  fisumcom2  11401  fprod2dlemstep  11585  fprodcom2fi  11589  eucalgval  12008  eucalginv  12010  eucalglt  12011  eucalg  12013  sqpweven  12129  2sqpwodd  12130  ctiunctlemudc  12392  tx2cn  13064  txdis  13071  txhmeo  13113  xmetxp  13301  xmetxpbl  13302  xmettxlem  13303  xmettx  13304