brfvrcld - Mathbox for Richard Penner

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Theorem brfvrcld 42277

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 42262	. . 3 ⊢ r* = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ {0, 1} (𝑟↑_𝑟𝑛))
2		brfvrcld.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
3		0nn0 12471	. . . . 5 ⊢ 0 ∈ ℕ₀
4		1nn0 12472	. . . . 5 ⊢ 1 ∈ ℕ₀
5		prssi 4818	. . . . 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 42269	. 2 ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ ∃𝑛 ∈ {0, 1}𝐴(𝑅↑_𝑟𝑛)𝐵))
9		oveq2 7402	. . . . 5 ⊢ (𝑛 = 0 → (𝑅↑_𝑟𝑛) = (𝑅↑_𝑟0))
10	9	breqd 5153	. . . 4 ⊢ (𝑛 = 0 → (𝐴(𝑅↑_𝑟𝑛)𝐵 ↔ 𝐴(𝑅↑_𝑟0)𝐵))
11		oveq2 7402	. . . . 5 ⊢ (𝑛 = 1 → (𝑅↑_𝑟𝑛) = (𝑅↑_𝑟1))
12	11	breqd 5153	. . . 4 ⊢ (𝑛 = 1 → (𝐴(𝑅↑_𝑟𝑛)𝐵 ↔ 𝐴(𝑅↑_𝑟1)𝐵))
13	10, 12	rexprg 4694	. . 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 1541 ∈ wcel 2106 ∃wrex 3070 Vcvv 3474 ⊆ wss 3945 {cpr 4625 class class class wbr 5142 ‘cfv 6533 (class class class)co 7394 0cc0 11094 1c1 11095 ℕ₀cn0 12456 ↑_𝑟crelexp 14950 r*crcl 42258

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7840 df-2nd 7960 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-er 8688 df-en 8925 df-dom 8926 df-sdom 8927 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-nn 12197 df-n0 12457 df-z 12543 df-uz 12807 df-seq 13951 df-relexp 14951 df-rcl 42259

This theorem is referenced by: brfvrcld2 42278

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