brfvrcld - Mathbox for Richard Penner

	Mathbox for Richard Penner		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > brfvrcld		Structured version Visualization version GIF version

Theorem brfvrcld 42745

Description: If two elements are connected by the reflexive closure of a relation, then they are connected via zero or one instances the relation. (Contributed by RP, 21-Jul-2020.)

**Hypothesis**
Ref	Expression
brfvrcld.r	⊢ (𝜑 → 𝑅 ∈ V)

**Assertion**
Ref	Expression
brfvrcld	⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ (𝐴(𝑅↑_𝑟0)𝐵 ∨ 𝐴(𝑅↑_𝑟1)𝐵)))

Proof of Theorem brfvrcld Dummy variables 𝑛 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfrcl4 42730	. . 3 ⊢ r* = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ {0, 1} (𝑟↑_𝑟𝑛))
2		brfvrcld.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
3		0nn0 12492	. . . . 5 ⊢ 0 ∈ ℕ₀
4		1nn0 12493	. . . . 5 ⊢ 1 ∈ ℕ₀
5		prssi 4824	. . . . 5 ⊢ ((0 ∈ ℕ₀ ∧ 1 ∈ ℕ₀) → {0, 1} ⊆ ℕ₀)
6	3, 4, 5	mp2an 689	. . . 4 ⊢ {0, 1} ⊆ ℕ₀
7	6	a1i 11	. . 3 ⊢ (𝜑 → {0, 1} ⊆ ℕ₀)
8	1, 2, 7	brmptiunrelexpd 42737	. 2 ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ ∃𝑛 ∈ {0, 1}𝐴(𝑅↑_𝑟𝑛)𝐵))
9		oveq2 7420	. . . . 5 ⊢ (𝑛 = 0 → (𝑅↑_𝑟𝑛) = (𝑅↑_𝑟0))
10	9	breqd 5159	. . . 4 ⊢ (𝑛 = 0 → (𝐴(𝑅↑_𝑟𝑛)𝐵 ↔ 𝐴(𝑅↑_𝑟0)𝐵))
11		oveq2 7420	. . . . 5 ⊢ (𝑛 = 1 → (𝑅↑_𝑟𝑛) = (𝑅↑_𝑟1))
12	11	breqd 5159	. . . 4 ⊢ (𝑛 = 1 → (𝐴(𝑅↑_𝑟𝑛)𝐵 ↔ 𝐴(𝑅↑_𝑟1)𝐵))
13	10, 12	rexprg 4700	. . 3 ⊢ ((0 ∈ ℕ₀ ∧ 1 ∈ ℕ₀) → (∃𝑛 ∈ {0, 1}𝐴(𝑅↑_𝑟𝑛)𝐵 ↔ (𝐴(𝑅↑_𝑟0)𝐵 ∨ 𝐴(𝑅↑_𝑟1)𝐵)))
14	3, 4, 13	mp2an 689	. 2 ⊢ (∃𝑛 ∈ {0, 1}𝐴(𝑅↑_𝑟𝑛)𝐵 ↔ (𝐴(𝑅↑_𝑟0)𝐵 ∨ 𝐴(𝑅↑_𝑟1)𝐵))
15	8, 14	bitrdi 287	1 ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ (𝐴(𝑅↑_𝑟0)𝐵 ∨ 𝐴(𝑅↑_𝑟1)𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∨ wo 844 = wceq 1540 ∈ wcel 2105 ∃wrex 3069 Vcvv 3473 ⊆ wss 3948 {cpr 4630 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 0cc0 11113 1c1 11114 ℕ₀cn0 12477 ↑_𝑟crelexp 14971 r*crcl 42726

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-n0 12478 df-z 12564 df-uz 12828 df-seq 13972 df-relexp 14972 df-rcl 42727

This theorem is referenced by: brfvrcld2 42746

W3C validator