brfvrcld - Mathbox for Richard Penner

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Theorem brfvrcld 42201

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 42186	. . 3 ⊢ r* = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ {0, 1} (𝑟↑_𝑟𝑛))
2		brfvrcld.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
3		0nn0 12468	. . . . 5 ⊢ 0 ∈ ℕ₀
4		1nn0 12469	. . . . 5 ⊢ 1 ∈ ℕ₀
5		prssi 4816	. . . . 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 42193	. 2 ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ ∃𝑛 ∈ {0, 1}𝐴(𝑅↑_𝑟𝑛)𝐵))
9		oveq2 7400	. . . . 5 ⊢ (𝑛 = 0 → (𝑅↑_𝑟𝑛) = (𝑅↑_𝑟0))
10	9	breqd 5151	. . . 4 ⊢ (𝑛 = 0 → (𝐴(𝑅↑_𝑟𝑛)𝐵 ↔ 𝐴(𝑅↑_𝑟0)𝐵))
11		oveq2 7400	. . . . 5 ⊢ (𝑛 = 1 → (𝑅↑_𝑟𝑛) = (𝑅↑_𝑟1))
12	11	breqd 5151	. . . 4 ⊢ (𝑛 = 1 → (𝐴(𝑅↑_𝑟𝑛)𝐵 ↔ 𝐴(𝑅↑_𝑟1)𝐵))
13	10, 12	rexprg 4692	. . 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 3069 Vcvv 3472 ⊆ wss 3943 {cpr 4623 class class class wbr 5140 ‘cfv 6531 (class class class)co 7392 0cc0 11091 1c1 11092 ℕ₀cn0 12453 ↑_𝑟crelexp 14947 r*crcl 42182

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 2702 ax-rep 5277 ax-sep 5291 ax-nul 5298 ax-pow 5355 ax-pr 5419 ax-un 7707 ax-cnex 11147 ax-resscn 11148 ax-1cn 11149 ax-icn 11150 ax-addcl 11151 ax-addrcl 11152 ax-mulcl 11153 ax-mulrcl 11154 ax-mulcom 11155 ax-addass 11156 ax-mulass 11157 ax-distr 11158 ax-i2m1 11159 ax-1ne0 11160 ax-1rid 11161 ax-rnegex 11162 ax-rrecex 11163 ax-cnre 11164 ax-pre-lttri 11165 ax-pre-lttrn 11166 ax-pre-ltadd 11167 ax-pre-mulgt0 11168

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 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 3474 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4943 df-iun 4991 df-br 5141 df-opab 5203 df-mpt 5224 df-tr 5258 df-id 5566 df-eprel 5572 df-po 5580 df-so 5581 df-fr 5623 df-we 5625 df-xp 5674 df-rel 5675 df-cnv 5676 df-co 5677 df-dm 5678 df-rn 5679 df-res 5680 df-ima 5681 df-pred 6288 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7348 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7838 df-2nd 7957 df-frecs 8247 df-wrecs 8278 df-recs 8352 df-rdg 8391 df-er 8685 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11231 df-mnf 11232 df-xr 11233 df-ltxr 11234 df-le 11235 df-sub 11427 df-neg 11428 df-nn 12194 df-n0 12454 df-z 12540 df-uz 12804 df-seq 13948 df-relexp 14948 df-rcl 42183

This theorem is referenced by: brfvrcld2 42202

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