op2nd - Intuitionistic Logic Explorer

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Theorem op2nd 6385

Description: Extract the second 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
op2nd	⊢ (2^nd ‘⟨𝐴, 𝐵⟩) = 𝐵

**Proof of Theorem op2nd**
Step	Hyp	Ref	Expression
1		op1st.1	. . . 4 ⊢ 𝐴 ∈ V
2		op1st.2	. . . 4 ⊢ 𝐵 ∈ V
3		opexg 4370	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ V)
4	1, 2, 3	mp2an 430	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
5		2ndvalg 6381	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ V → (2^nd ‘⟨𝐴, 𝐵⟩) = ∪ ran {⟨𝐴, 𝐵⟩})
6	4, 5	ax-mp 5	. 2 ⊢ (2^nd ‘⟨𝐴, 𝐵⟩) = ∪ ran {⟨𝐴, 𝐵⟩}
7	1, 2	op2nda 5275	. 2 ⊢ ∪ ran {⟨𝐴, 𝐵⟩} = 𝐵
8	6, 7	eqtri 2259	1 ⊢ (2^nd ‘⟨𝐴, 𝐵⟩) = 𝐵

Colors of variables: wff set class

This proof depends on syntax axioms: = wceq 1402 ∈ wcel 2209 Vcvv 2821 {csn 3709 ⟨cop 3712 ∪ cuni 3935 ran crn 4777 ‘cfv 5380 2^nd c2nd 6377

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 4251 ax-pow 4313 ax-pr 4348 ax-un 4580

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 4132 df-opab 4194 df-mpt 4195 df-id 4440 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-iota 5340 df-fun 5382 df-fv 5388 df-2nd 6379