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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 8025 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) | |
2 | 1 | eqeq1d 2736 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((2nd ‘〈𝐴, 𝐵〉) = 𝐶 ↔ 𝐵 = 𝐶)) |
3 | fo2nd 8033 | . . . 4 ⊢ 2nd :V–onto→V | |
4 | fofn 6822 | . . . 4 ⊢ (2nd :V–onto→V → 2nd Fn V) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ 2nd Fn V |
6 | opex 5474 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
7 | fnbrfvb 6959 | . . 3 ⊢ ((2nd Fn V ∧ 〈𝐴, 𝐵〉 ∈ V) → ((2nd ‘〈𝐴, 𝐵〉) = 𝐶 ↔ 〈𝐴, 𝐵〉2nd 𝐶)) | |
8 | 5, 6, 7 | mp2an 692 | . 2 ⊢ ((2nd ‘〈𝐴, 𝐵〉) = 𝐶 ↔ 〈𝐴, 𝐵〉2nd 𝐶) |
9 | eqcom 2741 | . 2 ⊢ (𝐵 = 𝐶 ↔ 𝐶 = 𝐵) | |
10 | 2, 8, 9 | 3bitr3g 313 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉2nd 𝐶 ↔ 𝐶 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 〈cop 4636 class class class wbr 5147 Fn wfn 6557 –onto→wfo 6560 ‘cfv 6562 2nd c2nd 8011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fo 6568 df-fv 6570 df-2nd 8013 |
This theorem is referenced by: br2ndeq 35752 fv2ndcnv 35758 brxrn 38355 |
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