brfvrcld - Mathbox for Richard Penner

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Theorem brfvrcld 43124

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 43109	. . 3 ⊢ r* = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ {0, 1} (𝑟↑_𝑟𝑛))
2		brfvrcld.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
3		0nn0 12523	. . . . 5 ⊢ 0 ∈ ℕ₀
4		1nn0 12524	. . . . 5 ⊢ 1 ∈ ℕ₀
5		prssi 4827	. . . . 5 ⊢ ((0 ∈ ℕ₀ ∧ 1 ∈ ℕ₀) → {0, 1} ⊆ ℕ₀)
6	3, 4, 5	mp2an 690	. . . 4 ⊢ {0, 1} ⊆ ℕ₀
7	6	a1i 11	. . 3 ⊢ (𝜑 → {0, 1} ⊆ ℕ₀)
8	1, 2, 7	brmptiunrelexpd 43116	. 2 ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ ∃𝑛 ∈ {0, 1}𝐴(𝑅↑_𝑟𝑛)𝐵))
9		oveq2 7432	. . . . 5 ⊢ (𝑛 = 0 → (𝑅↑_𝑟𝑛) = (𝑅↑_𝑟0))
10	9	breqd 5161	. . . 4 ⊢ (𝑛 = 0 → (𝐴(𝑅↑_𝑟𝑛)𝐵 ↔ 𝐴(𝑅↑_𝑟0)𝐵))
11		oveq2 7432	. . . . 5 ⊢ (𝑛 = 1 → (𝑅↑_𝑟𝑛) = (𝑅↑_𝑟1))
12	11	breqd 5161	. . . 4 ⊢ (𝑛 = 1 → (𝐴(𝑅↑_𝑟𝑛)𝐵 ↔ 𝐴(𝑅↑_𝑟1)𝐵))
13	10, 12	rexprg 4703	. . 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 690	. 2 ⊢ (∃𝑛 ∈ {0, 1}𝐴(𝑅↑_𝑟𝑛)𝐵 ↔ (𝐴(𝑅↑_𝑟0)𝐵 ∨ 𝐴(𝑅↑_𝑟1)𝐵))
15	8, 14	bitrdi 286	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 845 = wceq 1533 ∈ wcel 2098 ∃wrex 3066 Vcvv 3471 ⊆ wss 3947 {cpr 4632 class class class wbr 5150 ‘cfv 6551 (class class class)co 7424 0cc0 11144 1c1 11145 ℕ₀cn0 12508 ↑_𝑟crelexp 15004 r*crcl 43105

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-n0 12509 df-z 12595 df-uz 12859 df-seq 14005 df-relexp 15005 df-rcl 43106

This theorem is referenced by: brfvrcld2 43125

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