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Theorem wdom2d2 38126
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 7111 . . 3 ((𝐵𝑊𝐶𝑋) → (𝐵 × 𝐶) ∈ V)
52, 3, 4syl2anc 573 . 2 (𝜑 → (𝐵 × 𝐶) ∈ V)
6 wdom2d2.o . . 3 ((𝜑𝑥𝐴) → ∃𝑦𝐵𝑧𝐶 𝑥 = 𝑋)
7 nfcsb1v 3698 . . . . 5 𝑦(1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋
87nfeq2 2929 . . . 4 𝑦 𝑥 = (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋
9 nfcv 2913 . . . . . 6 𝑧(1st𝑤)
10 nfcsb1v 3698 . . . . . 6 𝑧(2nd𝑤) / 𝑧𝑋
119, 10nfcsb 3700 . . . . 5 𝑧(1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋
1211nfeq2 2929 . . . 4 𝑧 𝑥 = (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋
13 nfv 1995 . . . 4 𝑤 𝑥 = 𝑋
14 csbopeq1a 7374 . . . . 5 (𝑤 = ⟨𝑦, 𝑧⟩ → (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋 = 𝑋)
1514eqeq2d 2781 . . . 4 (𝑤 = ⟨𝑦, 𝑧⟩ → (𝑥 = (1st𝑤) / 𝑦(2nd𝑤) / 𝑧𝑋𝑥 = 𝑋))
168, 12, 13, 15rexxpf 5407 . . 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 8645 1 (𝜑𝐴* (𝐵 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wrex 3062  Vcvv 3351  csb 3682  cop 4323   class class class wbr 4787   × cxp 5248  cfv 6030  1st c1st 7317  2nd c2nd 7318  * cwdom 8622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-1st 7319  df-2nd 7320  df-er 7900  df-en 8114  df-dom 8115  df-sdom 8116  df-wdom 8624
This theorem is referenced by: (None)
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