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Theorem wdom2d2 38121
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)
4 xpexg 7106 . . 3 ((𝐵𝑊𝐶𝑋) → (𝐵 × 𝐶) ∈ V)
52, 3, 4syl2anc 565 . 2 (𝜑 → (𝐵 × 𝐶) ∈ V)
6 wdom2d2.o . . 3 ((𝜑𝑥𝐴) → ∃𝑦𝐵𝑧𝐶 𝑥 = 𝑋)
7 nfcsb1v 3696 . . . . 5 𝑦(1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋
87nfeq2 2928 . . . 4 𝑦 𝑥 = (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋
9 nfcv 2912 . . . . . 6 𝑧(1st𝑤)
10 nfcsb1v 3696 . . . . . 6 𝑧(2nd𝑤) / 𝑧𝑋
119, 10nfcsb 3698 . . . . 5 𝑧(1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋
1211nfeq2 2928 . . . 4 𝑧 𝑥 = (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋
13 nfv 1994 . . . 4 𝑤 𝑥 = 𝑋
14 csbopeq1a 7369 . . . . 5 (𝑤 = ⟨𝑦, 𝑧⟩ → (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋 = 𝑋)
1514eqeq2d 2780 . . . 4 (𝑤 = ⟨𝑦, 𝑧⟩ → (𝑥 = (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋𝑥 = 𝑋))
168, 12, 13, 15rexxpf 5408 . . 3 (∃𝑤 ∈ (𝐵 × 𝐶)𝑥 = (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋 ↔ ∃𝑦𝐵𝑧𝐶 𝑥 = 𝑋)
176, 16sylibr 224 . 2 ((𝜑𝑥𝐴) → ∃𝑤 ∈ (𝐵 × 𝐶)𝑥 = (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋)
181, 5, 17wdom2d 8640 1 (𝜑𝐴* (𝐵 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  wrex 3061  Vcvv 3349  csb 3680  cop 4320   class class class wbr 4784   × cxp 5247  cfv 6031  1st c1st 7312  2nd c2nd 7313  * cwdom 8617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-1st 7314  df-2nd 7315  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-wdom 8619
This theorem is referenced by: (None)
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