Theorem wdom2d2 43074

Description: Deduction for weak dominance by a Cartesian product. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)

Hypotheses

Ref	Expression
wdom2d2.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
wdom2d2.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
wdom2d2.c	⊢ (𝜑 → 𝐶 ∈ 𝑋)
wdom2d2.o	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = 𝑋)

Assertion

Ref	Expression
wdom2d2	⊢ (𝜑 → 𝐴 ≼* (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.

Step	Hyp	Ref	Expression
1		wdom2d2.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		wdom2d2.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3		wdom2d2.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
4	2, 3	xpexd 7684	. 2 ⊢ (𝜑 → (𝐵 × 𝐶) ∈ V)
5		wdom2d2.o	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = 𝑋)
6		nfcsb1v 3874	. . . . 5 ⊢ Ⅎ𝑦⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋
7	6	nfeq2 2912	. . . 4 ⊢ Ⅎ𝑦 𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋
8		nfcv 2894	. . . . . 6 ⊢ Ⅎ𝑧(1st ‘𝑤)
9		nfcsb1v 3874	. . . . . 6 ⊢ Ⅎ𝑧⦋(2nd ‘𝑤) / 𝑧⦌𝑋
10	8, 9	nfcsbw 3876	. . . . 5 ⊢ Ⅎ𝑧⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋
11	10	nfeq2 2912	. . . 4 ⊢ Ⅎ𝑧 𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋
12		nfv 1915	. . . 4 ⊢ Ⅎ𝑤 𝑥 = 𝑋
13		csbopeq1a 7982	. . . . 5 ⊢ (𝑤 = ⟨𝑦, 𝑧⟩ → ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 = 𝑋)
14	13	eqeq2d 2742	. . . 4 ⊢ (𝑤 = ⟨𝑦, 𝑧⟩ → (𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 ↔ 𝑥 = 𝑋))
15	7, 11, 12, 14	rexxpf 5787	. . 3 ⊢ (∃𝑤 ∈ (𝐵 × 𝐶)𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = 𝑋)
16	5, 15	sylibr 234	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑤 ∈ (𝐵 × 𝐶)𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋)
17	1, 4, 16	wdom2d 9466	1 ⊢ (𝜑 → 𝐴 ≼* (𝐵 × 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  Vcvv 3436  ⦋csb 3850  ⟨cop 4582   class class class wbr 5091   × cxp 5614  ‘cfv 6481  1^st c1st 7919  2^nd c2nd 7920   ≼^* cwdom 9450

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-1st 7921  df-2nd 7922  df-en 8870  df-dom 8871  df-sdom 8872  df-wdom 9451

This theorem is referenced by: (None)