brub - Mathbox for Scott Fenton

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Theorem brub 33415

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 5601	. . . . 5 ⊢ (𝑆(V × V)𝐴 ↔ (𝑆 ∈ V ∧ 𝐴 ∈ V))
4	1, 2, 3	mpbir2an 709	. . . 4 ⊢ 𝑆(V × V)𝐴
5		brdif 5119	. . . 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 32988	. . 3 ⊢ (𝑆((V ∖ 𝑅) ∘ ^◡ E )𝐴 ↔ ∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴)
8	6, 7	xchbinx 336	. 2 ⊢ (𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))𝐴 ↔ ¬ ∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴)
9		df-ub 33337	. . 3 ⊢ UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))
10	9	breqi 5072	. 2 ⊢ (𝑆UB𝑅𝐴 ↔ 𝑆((V × V) ∖ ((V ∖ 𝑅) ∘ ^◡ E ))𝐴)
11		brv 5364	. . . . . 6 ⊢ 𝑥V𝐴
12		brdif 5119	. . . . . 6 ⊢ (𝑥(V ∖ 𝑅)𝐴 ↔ (𝑥V𝐴 ∧ ¬ 𝑥𝑅𝐴))
13	11, 12	mpbiran 707	. . . . 5 ⊢ (𝑥(V ∖ 𝑅)𝐴 ↔ ¬ 𝑥𝑅𝐴)
14	13	rexbii 3247	. . . 4 ⊢ (∃𝑥 ∈ 𝑆 𝑥(V ∖ 𝑅)𝐴 ↔ ∃𝑥 ∈ 𝑆 ¬ 𝑥𝑅𝐴)
15		rexnal 3238	. . . 4 ⊢ (∃𝑥 ∈ 𝑆 ¬ 𝑥𝑅𝐴 ↔ ¬ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴)
16	14, 15	bitri 277	. . 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 305	1 ⊢ (𝑆UB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥𝑅𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 208 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ∖ cdif 3933 class class class wbr 5066 E cep 5464 × cxp 5553 ^◡ccnv 5554 ∘ ccom 5559 UBcub 33313

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-eprel 5465 df-xp 5561 df-cnv 5563 df-co 5564 df-ub 33337

This theorem is referenced by: brlb 33416

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