op1st - Intuitionistic Logic Explorer

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Theorem op1st 6380

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 4368	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ V)
4	1, 2, 3	mp2an 430	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
5		1stvalg 6376	. . 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 5269	. 2 ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = 𝐴
8	6, 7	eqtri 2259	1 ⊢ (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴

Colors of variables: wff set class

This proof depends on syntax axioms: = wceq 1402 ∈ wcel 2209 Vcvv 2821 {csn 3709 ⟨cop 3712 ∪ cuni 3935 dom cdm 4774 ‘cfv 5377 1^st c1st 6372

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ral 2533 df-rex 2534 df-v 2823 df-sbc 3052 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-br 4131 df-opab 4193 df-mpt 4194 df-id 4438 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-iota 5337 df-fun 5379 df-fv 5385 df-1st 6374