2ndrn - Intuitionistic Logic Explorer

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Theorem 2ndrn 5829

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 108	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ 𝑅)
2		1st2nd 5827	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
3	2, 1	eqeltrrd 2156	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ 𝑅)
4		1stexg 5814	. . . 4 ⊢ (𝐴 ∈ 𝑅 → (1^st ‘𝐴) ∈ V)
5		2ndexg 5815	. . . 4 ⊢ (𝐴 ∈ 𝑅 → (2^nd ‘𝐴) ∈ V)
6	4, 5	jca 300	. . 3 ⊢ (𝐴 ∈ 𝑅 → ((1^st ‘𝐴) ∈ V ∧ (2^nd ‘𝐴) ∈ V))
7		opelrng 4584	. . . 4 ⊢ (((1^st ‘𝐴) ∈ V ∧ (2^nd ‘𝐴) ∈ V ∧ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ 𝑅) → (2^nd ‘𝐴) ∈ ran 𝑅)
8	7	3expa 1138	. . 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 277	. 2 ⊢ ((𝐴 ∈ 𝑅 ∧ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ 𝑅) → (2^nd ‘𝐴) ∈ ran 𝑅)
10	1, 3, 9	syl2anc 403	1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (2^nd ‘𝐴) ∈ ran 𝑅)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 102 ∈ wcel 1433 Vcvv 2601 ⟨cop 3401 ran crn 4364 Rel wrel 4368 ‘cfv 4922 1^st c1st 5785 2^nd c2nd 5786

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fo 4928 df-fv 4930 df-1st 5787 df-2nd 5788

This theorem is referenced by: (None)