2ndrn - Intuitionistic Logic Explorer

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Theorem 2ndrn 6151

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 109	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ 𝑅)
2		1st2nd 6149	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
3	2, 1	eqeltrrd 2244	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ 𝑅)
4		1stexg 6135	. . . 4 ⊢ (𝐴 ∈ 𝑅 → (1^st ‘𝐴) ∈ V)
5		2ndexg 6136	. . . 4 ⊢ (𝐴 ∈ 𝑅 → (2^nd ‘𝐴) ∈ V)
6	4, 5	jca 304	. . 3 ⊢ (𝐴 ∈ 𝑅 → ((1^st ‘𝐴) ∈ V ∧ (2^nd ‘𝐴) ∈ V))
7		opelrng 4836	. . . 4 ⊢ (((1^st ‘𝐴) ∈ V ∧ (2^nd ‘𝐴) ∈ V ∧ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ 𝑅) → (2^nd ‘𝐴) ∈ ran 𝑅)
8	7	3expa 1193	. . 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 281	. 2 ⊢ ((𝐴 ∈ 𝑅 ∧ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ 𝑅) → (2^nd ‘𝐴) ∈ ran 𝑅)
10	1, 3, 9	syl2anc 409	1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (2^nd ‘𝐴) ∈ ran 𝑅)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2136 Vcvv 2726 ⟨cop 3579 ran crn 4605 Rel wrel 4609 ‘cfv 5188 1^st c1st 6106 2^nd c2nd 6107

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ral 2449 df-rex 2450 df-v 2728 df-sbc 2952 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-mpt 4045 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-fo 5194 df-fv 5196 df-1st 6108 df-2nd 6109

This theorem is referenced by: (None)