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Theorem wdom2d2 42237
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 7742 . 2 (𝜑 → (𝐵 × 𝐶) ∈ V)
5 wdom2d2.o . . 3 ((𝜑𝑥𝐴) → ∃𝑦𝐵𝑧𝐶 𝑥 = 𝑋)
6 nfcsb1v 3918 . . . . 5 𝑦(1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋
76nfeq2 2919 . . . 4 𝑦 𝑥 = (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋
8 nfcv 2902 . . . . . 6 𝑧(1st𝑤)
9 nfcsb1v 3918 . . . . . 6 𝑧(2nd𝑤) / 𝑧𝑋
108, 9nfcsbw 3920 . . . . 5 𝑧(1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋
1110nfeq2 2919 . . . 4 𝑧 𝑥 = (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋
12 nfv 1916 . . . 4 𝑤 𝑥 = 𝑋
13 csbopeq1a 8040 . . . . 5 (𝑤 = ⟨𝑦, 𝑧⟩ → (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋 = 𝑋)
1413eqeq2d 2742 . . . 4 (𝑤 = ⟨𝑦, 𝑧⟩ → (𝑥 = (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋𝑥 = 𝑋))
157, 11, 12, 14rexxpf 5847 . . 3 (∃𝑤 ∈ (𝐵 × 𝐶)𝑥 = (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋 ↔ ∃𝑦𝐵𝑧𝐶 𝑥 = 𝑋)
165, 15sylibr 233 . 2 ((𝜑𝑥𝐴) → ∃𝑤 ∈ (𝐵 × 𝐶)𝑥 = (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋)
171, 4, 16wdom2d 9581 1 (𝜑𝐴* (𝐵 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  wrex 3069  Vcvv 3473  csb 3893  cop 4634   class class class wbr 5148   × cxp 5674  cfv 6543  1st c1st 7977  2nd c2nd 7978  * cwdom 9565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-1st 7979  df-2nd 7980  df-en 8946  df-dom 8947  df-sdom 8948  df-wdom 9566
This theorem is referenced by: (None)
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