brub - Mathbox for Scott Fenton

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Theorem brub 32061

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 5147	. . . . 5 ⊢ (𝑆(V × V)𝐴 ↔ (𝑆 ∈ V ∧ 𝐴 ∈ V))
4	1, 2, 3	mpbir2an 955	. . . 4 ⊢ 𝑆(V × V)𝐴
5		brdif 4705	. . . 4 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))𝐴 ↔ (𝑆(V × V)𝐴 ∧ ¬ 𝑆((V ∖ 𝑅) ∘ ^◡ E )𝐴))
6	4, 5	mpbiran 953	. . 3 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))𝐴 ↔ ¬ 𝑆((V ∖ 𝑅) ∘ ^◡ E )𝐴)
7	1, 2	coepr 31642	. . 3 ⊢ (𝑆((V ∖ 𝑅) ∘ ^◡ E )𝐴 ↔ ∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴)
8	6, 7	xchbinx 324	. 2 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))𝐴 ↔ ¬ ∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴)
9		df-ub 31983	. . 3 ⊢ UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))
10	9	breqi 4659	. 2 ⊢ (𝑆UB𝑅𝐴 ↔ 𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))𝐴)
11		brv 4941	. . . . . 6 ⊢ 𝑥V𝐴
12		brdif 4705	. . . . . 6 ⊢ (𝑥(V ∖ 𝑅)𝐴 ↔ (𝑥V𝐴 ∧ ¬ 𝑥𝑅𝐴))
13	11, 12	mpbiran 953	. . . . 5 ⊢ (𝑥(V ∖ 𝑅)𝐴 ↔ ¬ 𝑥𝑅𝐴)
14	13	rexbii 3041	. . . 4 ⊢ (∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴 ↔ ∃𝑥 ∈ 𝑆 ¬ 𝑥𝑅𝐴)
15		rexnal 2995	. . . 4 ⊢ (∃𝑥 ∈ 𝑆 ¬ 𝑥𝑅𝐴 ↔ ¬ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴)
16	14, 15	bitri 264	. . 3 ⊢ (∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴 ↔ ¬ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴)
17	16	con2bii 347	. 2 ⊢ (∀𝑥 ∈ 𝑆 𝑥𝑅𝐴 ↔ ¬ ∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴)
18	8, 10, 17	3bitr4i 292	1 ⊢ (𝑆UB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 196 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ∖ cdif 3571 class class class wbr 4653 E cep 5028 × cxp 5112 ^◡ccnv 5113 ∘ ccom 5118 UBcub 31959

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-eprel 5029 df-xp 5120 df-cnv 5122 df-co 5123 df-ub 31983

This theorem is referenced by: brlb 32062

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