Theorem op1std 6247

Description: Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)

Hypotheses

Ref	Expression
op1st.1	⊢ 𝐴 ∈ V
op1st.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
op1std	⊢ (𝐶 = ⟨𝐴, 𝐵⟩ → (1st ‘𝐶) = 𝐴)

Proof of Theorem op1std

Step	Hyp	Ref	Expression
1		fveq2 5589	. 2 ⊢ (𝐶 = ⟨𝐴, 𝐵⟩ → (1st ‘𝐶) = (1st ‘⟨𝐴, 𝐵⟩))
2		op1st.1	. . 3 ⊢ 𝐴 ∈ V
3		op1st.2	. . 3 ⊢ 𝐵 ∈ V
4	2, 3	op1st 6245	. 2 ⊢ (1st ‘⟨𝐴, 𝐵⟩) = 𝐴
5	1, 4	eqtrdi 2255	1 ⊢ (𝐶 = ⟨𝐴, 𝐵⟩ → (1st ‘𝐶) = 𝐴)

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2177  Vcvv 2773  ⟨cop 3641  ‘cfv 5280  1^st c1st 6237

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 4170  ax-pow 4226  ax-pr 4261  ax-un 4488

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 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-iota 5241  df-fun 5282  df-fv 5288  df-1st 6239

This theorem is referenced by:  xp1st  6264  sbcopeq1a  6286  csbopeq1a  6287  eloprabi  6295  mpomptsx  6296  dmmpossx  6298  fmpox  6299  fmpoco  6315  df1st2  6318  xporderlem  6330  xpf1o  6956  fisumcom2  11824  fprodcom2fi  12012  txbas  14805  cnmpt1st  14835  txhmeo  14866  lgsquadlem1  15629  lgsquadlem2  15630