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Mirrors > Home > MPE Home > Th. List > ot3rdg | Structured version Visualization version GIF version |
Description: Extract the third member of an ordered triple. (See ot1stg 7936 comment.) (Contributed by NM, 3-Apr-2015.) |
Ref | Expression |
---|---|
ot3rdg | ⊢ (𝐶 ∈ 𝑉 → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ot 4596 | . . 3 ⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ | |
2 | 1 | fveq2i 6846 | . 2 ⊢ (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) |
3 | opex 5422 | . . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V | |
4 | op2ndg 7935 | . . 3 ⊢ ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ 𝑉) → (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = 𝐶) | |
5 | 3, 4 | mpan 689 | . 2 ⊢ (𝐶 ∈ 𝑉 → (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = 𝐶) |
6 | 2, 5 | eqtrid 2789 | 1 ⊢ (𝐶 ∈ 𝑉 → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3446 ⟨cop 4593 ⟨cotp 4595 ‘cfv 6497 2nd c2nd 7921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-ot 4596 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fv 6505 df-2nd 7923 |
This theorem is referenced by: oteqimp 7941 el2xptp0 7969 sbcoteq1a 7984 xpord3lem 8082 splval 14640 splcl 14641 ida2 17946 coa2 17956 mamufval 21737 msrval 34135 mapdhval 40190 hdmap1val 40264 |
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