op1st - Intuitionistic Logic Explorer

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Theorem op1st 6037

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 4145	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ V)
4	1, 2, 3	mp2an 422	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
5		1stvalg 6033	. . 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 5015	. 2 ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = 𝐴
8	6, 7	eqtri 2158	1 ⊢ (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴

Colors of variables: wff set class

Syntax hints: = wceq 1331 ∈ wcel 1480 Vcvv 2681 {csn 3522 ⟨cop 3525 ∪ cuni 3731 dom cdm 4534 ‘cfv 5118 1^st c1st 6029

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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-iota 5083 df-fun 5120 df-fv 5126 df-1st 6031