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Theorem wdom2d2 40560
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 7536 . 2 (𝜑 → (𝐵 × 𝐶) ∈ V)
5 wdom2d2.o . . 3 ((𝜑𝑥𝐴) → ∃𝑦𝐵𝑧𝐶 𝑥 = 𝑋)
6 nfcsb1v 3836 . . . . 5 𝑦(1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋
76nfeq2 2921 . . . 4 𝑦 𝑥 = (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋
8 nfcv 2904 . . . . . 6 𝑧(1st𝑤)
9 nfcsb1v 3836 . . . . . 6 𝑧(2nd𝑤) / 𝑧𝑋
108, 9nfcsbw 3838 . . . . 5 𝑧(1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋
1110nfeq2 2921 . . . 4 𝑧 𝑥 = (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋
12 nfv 1922 . . . 4 𝑤 𝑥 = 𝑋
13 csbopeq1a 7821 . . . . 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 5716 . . 3 (∃𝑤 ∈ (𝐵 × 𝐶)𝑥 = (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋 ↔ ∃𝑦𝐵𝑧𝐶 𝑥 = 𝑋)
165, 15sylibr 237 . 2 ((𝜑𝑥𝐴) → ∃𝑤 ∈ (𝐵 × 𝐶)𝑥 = (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋)
171, 4, 16wdom2d 9196 1 (𝜑𝐴* (𝐵 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  wrex 3062  Vcvv 3408  csb 3811  cop 4547   class class class wbr 5053   × cxp 5549  cfv 6380  1st c1st 7759  2nd c2nd 7760  * cwdom 9180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-1st 7761  df-2nd 7762  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-wdom 9181
This theorem is referenced by: (None)
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