Theorem briunov2 38941

Description: Two classes related by the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the two classes are related by that operator value. (Contributed by RP, 1-Jun-2020.)

Hypothesis

Ref	Expression
briunov2.def	⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))

Assertion

Ref	Expression
briunov2	⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ 𝑁 𝑋(𝑅 ↑ 𝑛)𝑌))

Distinct variable groups:   𝑛,𝑟,𝐶,𝑁, ↑   𝑅,𝑛,𝑟   𝑛,𝑋   𝑛,𝑌

Allowed substitution hints:   𝑈(𝑛,𝑟)   𝑉(𝑛,𝑟)   𝑋(𝑟)   𝑌(𝑟)

Proof of Theorem briunov2

Step	Hyp	Ref	Expression
1		briunov2.def	. . 3 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))
2	1	eliunov2 38938	. 2 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (⟨𝑋, 𝑌⟩ ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 ⟨𝑋, 𝑌⟩ ∈ (𝑅 ↑ 𝑛)))
3		df-br 4889	. 2 ⊢ (𝑋(𝐶‘𝑅)𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ (𝐶‘𝑅))
4		df-br 4889	. . 3 ⊢ (𝑋(𝑅 ↑ 𝑛)𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ (𝑅 ↑ 𝑛))
5	4	rexbii 3224	. 2 ⊢ (∃𝑛 ∈ 𝑁 𝑋(𝑅 ↑ 𝑛)𝑌 ↔ ∃𝑛 ∈ 𝑁 ⟨𝑋, 𝑌⟩ ∈ (𝑅 ↑ 𝑛))
6	2, 3, 5	3bitr4g 306	1 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ 𝑁 𝑋(𝑅 ↑ 𝑛)𝑌))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∃wrex 3091  Vcvv 3398  ⟨cop 4404  ∪ ciun 4755   class class class wbr 4888   ↦ cmpt 4967  ‘cfv 6137  (class class class)co 6924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927

This theorem is referenced by:  brmptiunrelexpd  38942  brtrclrec  38955  brrtrclrec  38956  briunov2uz  38957