brub - Mathbox for Scott Fenton

	Mathbox for Scott Fenton		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > brub		Structured version Visualization version GIF version

Theorem brub 34914

Description: Binary relation form of the upper bound functor. (Contributed by Scott Fenton, 3-May-2018.)

**Hypotheses**
Ref	Expression
brub.1	⊢ 𝑆 ∈ V
brub.2	⊢ 𝐴 ∈ V

**Assertion**
Ref	Expression
brub	⊢ (𝑆UB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴)

Distinct variable groups: 𝑥,𝐴 𝑥,𝑅 𝑥,𝑆

**Proof of Theorem brub**
Step	Hyp	Ref	Expression
1		brub.1	. . . . 5 ⊢ 𝑆 ∈ V
2		brub.2	. . . . 5 ⊢ 𝐴 ∈ V
3		brxp 5723	. . . . 5 ⊢ (𝑆(V × V)𝐴 ↔ (𝑆 ∈ V ∧ 𝐴 ∈ V))
4	1, 2, 3	mpbir2an 709	. . . 4 ⊢ 𝑆(V × V)𝐴
5		brdif 5200	. . . 4 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))𝐴 ↔ (𝑆(V × V)𝐴 ∧ ¬ 𝑆((V ∖ 𝑅) ∘ ^◡ E )𝐴))
6	4, 5	mpbiran 707	. . 3 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))𝐴 ↔ ¬ 𝑆((V ∖ 𝑅) ∘ ^◡ E )𝐴)
7	1, 2	coepr 34711	. . 3 ⊢ (𝑆((V ∖ 𝑅) ∘ ^◡ E )𝐴 ↔ ∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴)
8	6, 7	xchbinx 333	. 2 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))𝐴 ↔ ¬ ∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴)
9		df-ub 34836	. . 3 ⊢ UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))
10	9	breqi 5153	. 2 ⊢ (𝑆UB𝑅𝐴 ↔ 𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))𝐴)
11		brv 5471	. . . . . 6 ⊢ 𝑥V𝐴
12		brdif 5200	. . . . . 6 ⊢ (𝑥(V ∖ 𝑅)𝐴 ↔ (𝑥V𝐴 ∧ ¬ 𝑥𝑅𝐴))
13	11, 12	mpbiran 707	. . . . 5 ⊢ (𝑥(V ∖ 𝑅)𝐴 ↔ ¬ 𝑥𝑅𝐴)
14	13	rexbii 3094	. . . 4 ⊢ (∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴 ↔ ∃𝑥 ∈ 𝑆 ¬ 𝑥𝑅𝐴)
15		rexnal 3100	. . . 4 ⊢ (∃𝑥 ∈ 𝑆 ¬ 𝑥𝑅𝐴 ↔ ¬ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴)
16	14, 15	bitri 274	. . 3 ⊢ (∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴 ↔ ¬ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴)
17	16	con2bii 357	. 2 ⊢ (∀𝑥 ∈ 𝑆 𝑥𝑅𝐴 ↔ ¬ ∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴)
18	8, 10, 17	3bitr4i 302	1 ⊢ (𝑆UB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 205 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ∖ cdif 3944 class class class wbr 5147 E cep 5578 × cxp 5673 ^◡ccnv 5674 ∘ ccom 5679 UBcub 34812

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-eprel 5579 df-xp 5681 df-cnv 5683 df-co 5684 df-ub 34836

This theorem is referenced by: brlb 34915

W3C validator