op1st - Intuitionistic Logic Explorer

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Theorem op1st 5998

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 4110	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ V)
4	1, 2, 3	mp2an 420	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
5		1stvalg 5994	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ V → (1^st ‘⟨𝐴, 𝐵⟩) = ∪ dom {⟨𝐴, 𝐵⟩})
6	4, 5	ax-mp 7	. 2 ⊢ (1^st ‘⟨𝐴, 𝐵⟩) = ∪ dom {⟨𝐴, 𝐵⟩}
7	1, 2	op1sta 4978	. 2 ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = 𝐴
8	6, 7	eqtri 2135	1 ⊢ (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴

Colors of variables: wff set class

Syntax hints: = wceq 1314 ∈ wcel 1463 Vcvv 2657 {csn 3493 ⟨cop 3496 ∪ cuni 3702 dom cdm 4499 ‘cfv 5081 1^st c1st 5990

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315

This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-v 2659 df-sbc 2879 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-mpt 3951 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-iota 5046 df-fun 5083 df-fv 5089 df-1st 5992