2ndrn - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > 2ndrn		GIF version

Theorem 2ndrn 6417

Description: The second ordered pair component of a member of a relation belongs to the range of the relation. (Contributed by NM, 17-Sep-2006.)

**Assertion**
Ref	Expression
2ndrn	⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (2^nd ‘𝐴) ∈ ran 𝑅)

**Proof of Theorem 2ndrn**
Step	Hyp	Ref	Expression
1		simpr 110	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ 𝑅)
2		1st2nd 6415	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
3	2, 1	eqeltrrd 2316	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ 𝑅)
4		1stexg 6401	. . . 4 ⊢ (𝐴 ∈ 𝑅 → (1^st ‘𝐴) ∈ V)
5		2ndexg 6402	. . . 4 ⊢ (𝐴 ∈ 𝑅 → (2^nd ‘𝐴) ∈ V)
6	4, 5	jca 306	. . 3 ⊢ (𝐴 ∈ 𝑅 → ((1^st ‘𝐴) ∈ V ∧ (2^nd ‘𝐴) ∈ V))
7		opelrng 5014	. . . 4 ⊢ (((1^st ‘𝐴) ∈ V ∧ (2^nd ‘𝐴) ∈ V ∧ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ 𝑅) → (2^nd ‘𝐴) ∈ ran 𝑅)
8	7	3expa 1234	. . 3 ⊢ ((((1^st ‘𝐴) ∈ V ∧ (2^nd ‘𝐴) ∈ V) ∧ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ 𝑅) → (2^nd ‘𝐴) ∈ ran 𝑅)
9	6, 8	sylan 283	. 2 ⊢ ((𝐴 ∈ 𝑅 ∧ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ 𝑅) → (2^nd ‘𝐴) ∈ ran 𝑅)
10	1, 3, 9	syl2anc 415	1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (2^nd ‘𝐴) ∈ ran 𝑅)

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ∈ wcel 2209 Vcvv 2821 ⟨cop 3712 ran crn 4775 Rel wrel 4779 ‘cfv 5377 1^st c1st 6372 2^nd c2nd 6373

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-fn 5380 df-f 5381 df-fo 5383 df-fv 5385 df-1st 6374 df-2nd 6375

This theorem is used by: (None)

W3C validator