brub - Mathbox for Scott Fenton

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Theorem brub 36454

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 5709	. . . . 5 ⊢ (𝑆(V × V)𝐴 ↔ (𝑆 ∈ V ∧ 𝐴 ∈ V))
4	1, 2, 3	mpbir2an 723	. . . 4 ⊢ 𝑆(V × V)𝐴
5		brdif 5163	. . . 4 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))𝐴 ↔ (𝑆(V × V)𝐴 ∧ ¬ 𝑆((V ∖ 𝑅) ∘ ^◡ E )𝐴))
6	4, 5	mpbiran 721	. . 3 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))𝐴 ↔ ¬ 𝑆((V ∖ 𝑅) ∘ ^◡ E )𝐴)
7	1, 2	coepr 36253	. . 3 ⊢ (𝑆((V ∖ 𝑅) ∘ ^◡ E )𝐴 ↔ ∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴)
8	6, 7	xchbinx 337	. 2 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))𝐴 ↔ ¬ ∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴)
9		df-ub 36374	. . 3 ⊢ UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))
10	9	breqi 5114	. 2 ⊢ (𝑆UB𝑅𝐴 ↔ 𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))𝐴)
11		brv 5453	. . . . . 6 ⊢ 𝑥V𝐴
12		brdif 5163	. . . . . 6 ⊢ (𝑥(V ∖ 𝑅)𝐴 ↔ (𝑥V𝐴 ∧ ¬ 𝑥𝑅𝐴))
13	11, 12	mpbiran 721	. . . . 5 ⊢ (𝑥(V ∖ 𝑅)𝐴 ↔ ¬ 𝑥𝑅𝐴)
14	13	rexbii 3111	. . . 4 ⊢ (∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴 ↔ ∃𝑥 ∈ 𝑆 ¬ 𝑥𝑅𝐴)
15		rexnal 3116	. . . 4 ⊢ (∃𝑥 ∈ 𝑆 ¬ 𝑥𝑅𝐴 ↔ ¬ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴)
16	14, 15	bitri 278	. . 3 ⊢ (∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴 ↔ ¬ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴)
17	16	con2bii 360	. 2 ⊢ (∀𝑥 ∈ 𝑆 𝑥𝑅𝐴 ↔ ¬ ∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴)
18	8, 10, 17	3bitr4i 306	1 ⊢ (𝑆UB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 ↔ wb 209 ∈ wcel 2142 ∀wral 3078 ∃wrex 3088 Vcvv 3454 ∖ cdif 3901 class class class wbr 5108 E cep 5559 × cxp 5658 ^◡ccnv 5659 ∘ ccom 5664 UBcub 36350

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-ext 2734 ax-sep 5256 ax-pr 5403

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-sb 2096 df-clab 2741 df-cleq 2754 df-clel 2837 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3416 df-v 3456 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-sn 4589 df-pr 4591 df-op 4595 df-br 5109 df-opab 5173 df-eprel 5560 df-xp 5666 df-cnv 5668 df-co 5669 df-ub 36374

This theorem is used by: brlb 36455

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