op1st - Intuitionistic Logic Explorer

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Theorem op1st 6199

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 4257	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ V)
4	1, 2, 3	mp2an 426	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
5		1stvalg 6195	. . 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 5147	. 2 ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = 𝐴
8	6, 7	eqtri 2214	1 ⊢ (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴

Colors of variables: wff set class

Syntax hints: = wceq 1364 ∈ wcel 2164 Vcvv 2760 {csn 3618 ⟨cop 3621 ∪ cuni 3835 dom cdm 4659 ‘cfv 5254 1^st c1st 6191

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-sbc 2986 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-iota 5215 df-fun 5256 df-fv 5262 df-1st 6193