op1st - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > op1st		GIF version

Theorem op1st 6318

Description: Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)

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

**Assertion**
Ref	Expression
op1st	⊢ (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴

**Proof of Theorem op1st**
Step	Hyp	Ref	Expression
1		op1st.1	. . . 4 ⊢ 𝐴 ∈ V
2		op1st.2	. . . 4 ⊢ 𝐵 ∈ V
3		opexg 4326	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ V)
4	1, 2, 3	mp2an 426	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
5		1stvalg 6314	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ V → (1^st ‘⟨𝐴, 𝐵⟩) = ∪ dom {⟨𝐴, 𝐵⟩})
6	4, 5	ax-mp 5	. 2 ⊢ (1^st ‘⟨𝐴, 𝐵⟩) = ∪ dom {⟨𝐴, 𝐵⟩}
7	1, 2	op1sta 5225	. 2 ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = 𝐴
8	6, 7	eqtri 2252	1 ⊢ (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴

Colors of variables: wff set class

Syntax hints: = wceq 1398 ∈ wcel 2202 Vcvv 2803 {csn 3673 ⟨cop 3676 ∪ cuni 3898 dom cdm 4731 ‘cfv 5333 1^st c1st 6310

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ral 2516 df-rex 2517 df-v 2805 df-sbc 3033 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-br 4094 df-opab 4156 df-mpt 4157 df-id 4396 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-iota 5293 df-fun 5335 df-fv 5341 df-1st 6312