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Mirrors > Home > MPE Home > Th. List > br2ndeqg | Structured version Visualization version GIF version |
Description: Uniqueness condition for the binary relation 2nd. (Contributed by Scott Fenton, 2-Jul-2020.) Revised to remove sethood hypothesis on 𝐶. (Revised by Peter Mazsa, 17-Jan-2022.) |
Ref | Expression |
---|---|
br2ndeqg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉2nd 𝐶 ↔ 𝐶 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | op2ndg 7415 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) | |
2 | 1 | eqeq1d 2802 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((2nd ‘〈𝐴, 𝐵〉) = 𝐶 ↔ 𝐵 = 𝐶)) |
3 | fo2nd 7423 | . . . 4 ⊢ 2nd :V–onto→V | |
4 | fofn 6334 | . . . 4 ⊢ (2nd :V–onto→V → 2nd Fn V) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ 2nd Fn V |
6 | opex 5124 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
7 | fnbrfvb 6461 | . . 3 ⊢ ((2nd Fn V ∧ 〈𝐴, 𝐵〉 ∈ V) → ((2nd ‘〈𝐴, 𝐵〉) = 𝐶 ↔ 〈𝐴, 𝐵〉2nd 𝐶)) | |
8 | 5, 6, 7 | mp2an 684 | . 2 ⊢ ((2nd ‘〈𝐴, 𝐵〉) = 𝐶 ↔ 〈𝐴, 𝐵〉2nd 𝐶) |
9 | eqcom 2807 | . 2 ⊢ (𝐵 = 𝐶 ↔ 𝐶 = 𝐵) | |
10 | 2, 8, 9 | 3bitr3g 305 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉2nd 𝐶 ↔ 𝐶 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 Vcvv 3386 〈cop 4375 class class class wbr 4844 Fn wfn 6097 –onto→wfo 6100 ‘cfv 6102 2nd c2nd 7401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3388 df-sbc 3635 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-br 4845 df-opab 4907 df-mpt 4924 df-id 5221 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-fo 6108 df-fv 6110 df-2nd 7403 |
This theorem is referenced by: br2ndeq 32184 fv2ndcnv 32192 brxrn 34629 |
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