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Theorem brub 35955

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 5734	. . . . 5 ⊢ (𝑆(V × V)𝐴 ↔ (𝑆 ∈ V ∧ 𝐴 ∈ V))
4	1, 2, 3	mpbir2an 711	. . . 4 ⊢ 𝑆(V × V)𝐴
5		brdif 5196	. . . 4 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))𝐴 ↔ (𝑆(V × V)𝐴 ∧ ¬ 𝑆((V ∖ 𝑅) ∘ ^◡ E )𝐴))
6	4, 5	mpbiran 709	. . 3 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))𝐴 ↔ ¬ 𝑆((V ∖ 𝑅) ∘ ^◡ E )𝐴)
7	1, 2	coepr 35753	. . 3 ⊢ (𝑆((V ∖ 𝑅) ∘ ^◡ E )𝐴 ↔ ∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴)
8	6, 7	xchbinx 334	. 2 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))𝐴 ↔ ¬ ∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴)
9		df-ub 35877	. . 3 ⊢ UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))
10	9	breqi 5149	. 2 ⊢ (𝑆UB𝑅𝐴 ↔ 𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))𝐴)
11		brv 5477	. . . . . 6 ⊢ 𝑥V𝐴
12		brdif 5196	. . . . . 6 ⊢ (𝑥(V ∖ 𝑅)𝐴 ↔ (𝑥V𝐴 ∧ ¬ 𝑥𝑅𝐴))
13	11, 12	mpbiran 709	. . . . 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 275	. . 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 303	1 ⊢ (𝑆UB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 206 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ∖ cdif 3948 class class class wbr 5143 E cep 5583 × cxp 5683 ^◡ccnv 5684 ∘ ccom 5689 UBcub 35853

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-eprel 5584 df-xp 5691 df-cnv 5693 df-co 5694 df-ub 35877

This theorem is referenced by: brlb 35956

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