brfvrcld - Mathbox for Richard Penner

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Theorem brfvrcld 43681

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 43666	. . 3 ⊢ r* = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ {0, 1} (𝑟↑_𝑟𝑛))
2		brfvrcld.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
3		0nn0 12539	. . . . 5 ⊢ 0 ∈ ℕ₀
4		1nn0 12540	. . . . 5 ⊢ 1 ∈ ℕ₀
5		prssi 4826	. . . . 5 ⊢ ((0 ∈ ℕ₀ ∧ 1 ∈ ℕ₀) → {0, 1} ⊆ ℕ₀)
6	3, 4, 5	mp2an 692	. . . 4 ⊢ {0, 1} ⊆ ℕ₀
7	6	a1i 11	. . 3 ⊢ (𝜑 → {0, 1} ⊆ ℕ₀)
8	1, 2, 7	brmptiunrelexpd 43673	. 2 ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ ∃𝑛 ∈ {0, 1}𝐴(𝑅↑_𝑟𝑛)𝐵))
9		oveq2 7439	. . . . 5 ⊢ (𝑛 = 0 → (𝑅↑_𝑟𝑛) = (𝑅↑_𝑟0))
10	9	breqd 5159	. . . 4 ⊢ (𝑛 = 0 → (𝐴(𝑅↑_𝑟𝑛)𝐵 ↔ 𝐴(𝑅↑_𝑟0)𝐵))
11		oveq2 7439	. . . . 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 4702	. . 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 692	. 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 206 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 Vcvv 3478 ⊆ wss 3963 {cpr 4633 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 ℕ₀cn0 12524 ↑_𝑟crelexp 15055 r*crcl 43662

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-seq 14040 df-relexp 15056 df-rcl 43663

This theorem is referenced by: brfvrcld2 43682

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