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Mirrors > Home > MPE Home > Th. List > br1steqg | Structured version Visualization version GIF version |
Description: Uniqueness condition for the binary relation 1st. (Contributed by Scott Fenton, 2-Jul-2020.) Revised to remove sethood hypothesis on 𝐶. (Revised by Peter Mazsa, 17-Jan-2022.) |
Ref | Expression |
---|---|
br1steqg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉1st 𝐶 ↔ 𝐶 = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | op1stg 7690 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘〈𝐴, 𝐵〉) = 𝐴) | |
2 | 1 | eqeq1d 2820 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((1st ‘〈𝐴, 𝐵〉) = 𝐶 ↔ 𝐴 = 𝐶)) |
3 | fo1st 7698 | . . . 4 ⊢ 1st :V–onto→V | |
4 | fofn 6585 | . . . 4 ⊢ (1st :V–onto→V → 1st Fn V) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ 1st Fn V |
6 | opex 5347 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
7 | fnbrfvb 6711 | . . 3 ⊢ ((1st Fn V ∧ 〈𝐴, 𝐵〉 ∈ V) → ((1st ‘〈𝐴, 𝐵〉) = 𝐶 ↔ 〈𝐴, 𝐵〉1st 𝐶)) | |
8 | 5, 6, 7 | mp2an 688 | . 2 ⊢ ((1st ‘〈𝐴, 𝐵〉) = 𝐶 ↔ 〈𝐴, 𝐵〉1st 𝐶) |
9 | eqcom 2825 | . 2 ⊢ (𝐴 = 𝐶 ↔ 𝐶 = 𝐴) | |
10 | 2, 8, 9 | 3bitr3g 314 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉1st 𝐶 ↔ 𝐶 = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 〈cop 4563 class class class wbr 5057 Fn wfn 6343 –onto→wfo 6346 ‘cfv 6348 1st c1st 7676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-fv 6356 df-1st 7678 |
This theorem is referenced by: br1steq 32911 fv1stcnv 32917 brxrn 35506 |
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