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Theorem wdom2d2 43047
Description: Deduction for weak dominance by a Cartesian product. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
Hypotheses
Ref Expression
wdom2d2.a (𝜑𝐴𝑉)
wdom2d2.b (𝜑𝐵𝑊)
wdom2d2.c (𝜑𝐶𝑋)
wdom2d2.o ((𝜑𝑥𝐴) → ∃𝑦𝐵𝑧𝐶 𝑥 = 𝑋)
Assertion
Ref Expression
wdom2d2 (𝜑𝐴* (𝐵 × 𝐶))
Distinct variable groups:   𝑥,𝑋   𝑥,𝐴   𝑥,𝑦,𝐵   𝑥,𝑧,𝐶,𝑦   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐴(𝑦,𝑧)   𝐵(𝑧)   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)   𝑋(𝑦,𝑧)

Proof of Theorem wdom2d2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 wdom2d2.a . 2 (𝜑𝐴𝑉)
2 wdom2d2.b . . 3 (𝜑𝐵𝑊)
3 wdom2d2.c . . 3 (𝜑𝐶𝑋)
42, 3xpexd 7771 . 2 (𝜑 → (𝐵 × 𝐶) ∈ V)
5 wdom2d2.o . . 3 ((𝜑𝑥𝐴) → ∃𝑦𝐵𝑧𝐶 𝑥 = 𝑋)
6 nfcsb1v 3923 . . . . 5 𝑦(1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋
76nfeq2 2923 . . . 4 𝑦 𝑥 = (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋
8 nfcv 2905 . . . . . 6 𝑧(1st𝑤)
9 nfcsb1v 3923 . . . . . 6 𝑧(2nd𝑤) / 𝑧𝑋
108, 9nfcsbw 3925 . . . . 5 𝑧(1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋
1110nfeq2 2923 . . . 4 𝑧 𝑥 = (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋
12 nfv 1914 . . . 4 𝑤 𝑥 = 𝑋
13 csbopeq1a 8075 . . . . 5 (𝑤 = ⟨𝑦, 𝑧⟩ → (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋 = 𝑋)
1413eqeq2d 2748 . . . 4 (𝑤 = ⟨𝑦, 𝑧⟩ → (𝑥 = (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋𝑥 = 𝑋))
157, 11, 12, 14rexxpf 5858 . . 3 (∃𝑤 ∈ (𝐵 × 𝐶)𝑥 = (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋 ↔ ∃𝑦𝐵𝑧𝐶 𝑥 = 𝑋)
165, 15sylibr 234 . 2 ((𝜑𝑥𝐴) → ∃𝑤 ∈ (𝐵 × 𝐶)𝑥 = (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋)
171, 4, 16wdom2d 9620 1 (𝜑𝐴* (𝐵 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  csb 3899  cop 4632   class class class wbr 5143   × cxp 5683  cfv 6561  1st c1st 8012  2nd c2nd 8013  * cwdom 9604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1st 8014  df-2nd 8015  df-en 8986  df-dom 8987  df-sdom 8988  df-wdom 9605
This theorem is referenced by: (None)
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