Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > br1steq | Structured version Visualization version GIF version |
Description: Uniqueness condition for binary relationship over the 1st relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.) |
Ref | Expression |
---|---|
br1steq.1 | ⊢ 𝐴 ∈ V |
br1steq.2 | ⊢ 𝐵 ∈ V |
br1steq.3 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
br1steq | ⊢ (〈𝐴, 𝐵〉1st 𝐶 ↔ 𝐶 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | br1steq.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | br1steq.2 | . . . 4 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | op1st 7067 | . . 3 ⊢ (1st ‘〈𝐴, 𝐵〉) = 𝐴 |
4 | 3 | eqeq1i 2615 | . 2 ⊢ ((1st ‘〈𝐴, 𝐵〉) = 𝐶 ↔ 𝐴 = 𝐶) |
5 | fo1st 7079 | . . . 4 ⊢ 1st :V–onto→V | |
6 | fofn 6030 | . . . 4 ⊢ (1st :V–onto→V → 1st Fn V) | |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ 1st Fn V |
8 | opex 4859 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
9 | fnbrfvb 6146 | . . 3 ⊢ ((1st Fn V ∧ 〈𝐴, 𝐵〉 ∈ V) → ((1st ‘〈𝐴, 𝐵〉) = 𝐶 ↔ 〈𝐴, 𝐵〉1st 𝐶)) | |
10 | 7, 8, 9 | mp2an 704 | . 2 ⊢ ((1st ‘〈𝐴, 𝐵〉) = 𝐶 ↔ 〈𝐴, 𝐵〉1st 𝐶) |
11 | eqcom 2617 | . 2 ⊢ (𝐴 = 𝐶 ↔ 𝐶 = 𝐴) | |
12 | 4, 10, 11 | 3bitr3i 289 | 1 ⊢ (〈𝐴, 𝐵〉1st 𝐶 ↔ 𝐶 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 class class class wbr 4583 Fn wfn 5799 –onto→wfo 5802 ‘cfv 5804 1st c1st 7057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-1st 7059 |
This theorem is referenced by: br1steqg 30919 dfdm5 30921 brtxp 31157 brpprod 31162 elfuns 31192 brimg 31214 brcup 31216 brcap 31217 brrestrict 31226 |
Copyright terms: Public domain | W3C validator |