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Theorem prdsbasex 16423
Description: Lemma for structure products. (Contributed by Mario Carneiro, 3-Jan-2015.)
Hypothesis
Ref Expression
prdsbasex.b 𝐵 = X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))
Assertion
Ref Expression
prdsbasex 𝐵 ∈ V
Distinct variable group:   𝑥,𝑅
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem prdsbasex
StepHypRef Expression
1 prdsbasex.b . 2 𝐵 = X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))
2 ixpexg 8171 . . 3 (∀𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ∈ V → X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ∈ V)
3 fvexd 6425 . . 3 (𝑥 ∈ dom 𝑅 → (Base‘(𝑅𝑥)) ∈ V)
42, 3mprg 3106 . 2 X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ∈ V
51, 4eqeltri 2873 1 𝐵 ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  wcel 2157  Vcvv 3384  dom cdm 5311  cfv 6100  Xcixp 8147  Basecbs 16181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-rep 4963  ax-sep 4974  ax-nul 4982  ax-pow 5034  ax-pr 5096  ax-un 7182
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3386  df-sbc 3633  df-csb 3728  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4628  df-iun 4711  df-br 4843  df-opab 4905  df-mpt 4922  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6063  df-fun 6102  df-fn 6103  df-f 6104  df-f1 6105  df-fo 6106  df-f1o 6107  df-fv 6108  df-ixp 8148
This theorem is referenced by: (None)
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