2ndrn - Intuitionistic Logic Explorer

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Theorem 2ndrn 6250

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 6248	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
3	2, 1	eqeltrrd 2274	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ 𝑅)
4		1stexg 6234	. . . 4 ⊢ (𝐴 ∈ 𝑅 → (1^st ‘𝐴) ∈ V)
5		2ndexg 6235	. . . 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 4899	. . . 4 ⊢ (((1^st ‘𝐴) ∈ V ∧ (2^nd ‘𝐴) ∈ V ∧ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ 𝑅) → (2^nd ‘𝐴) ∈ ran 𝑅)
8	7	3expa 1205	. . 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 411	1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (2^nd ‘𝐴) ∈ ran 𝑅)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2167 Vcvv 2763 ⟨cop 3626 ran crn 4665 Rel wrel 4669 ‘cfv 5259 1^st c1st 6205 2^nd c2nd 6206

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ral 2480 df-rex 2481 df-v 2765 df-sbc 2990 df-un 3161 df-in 3163 df-ss 3170 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-br 4035 df-opab 4096 df-mpt 4097 df-id 4329 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-fo 5265 df-fv 5267 df-1st 6207 df-2nd 6208

This theorem is referenced by: (None)