Theorem wdom2d2 43481

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 7698	. 2 ⊢ (𝜑 → (𝐵 × 𝐶) ∈ V)
5		wdom2d2.o	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = 𝑋)
6		nfcsb1v 3862	. . . . 5 ⊢ Ⅎ𝑦⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋
7	6	nfeq2 2917	. . . 4 ⊢ Ⅎ𝑦 𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋
8		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑧(1st ‘𝑤)
9		nfcsb1v 3862	. . . . . 6 ⊢ Ⅎ𝑧⦋(2nd ‘𝑤) / 𝑧⦌𝑋
10	8, 9	nfcsbw 3864	. . . . 5 ⊢ Ⅎ𝑧⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋
11	10	nfeq2 2917	. . . 4 ⊢ Ⅎ𝑧 𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋
12		nfv 1916	. . . 4 ⊢ Ⅎ𝑤 𝑥 = 𝑋
13		csbopeq1a 7996	. . . . 5 ⊢ (𝑤 = ⟨𝑦, 𝑧⟩ → ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 = 𝑋)
14	13	eqeq2d 2748	. . . 4 ⊢ (𝑤 = ⟨𝑦, 𝑧⟩ → (𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 ↔ 𝑥 = 𝑋))
15	7, 11, 12, 14	rexxpf 5796	. . 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 9488	1 ⊢ (𝜑 → 𝐴 ≼* (𝐵 × 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  Vcvv 3430  ⦋csb 3838  ⟨cop 4574   class class class wbr 5086   × cxp 5622  ‘cfv 6492  1^st c1st 7933  2^nd c2nd 7934   ≼^* cwdom 9472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-1st 7935  df-2nd 7936  df-en 8887  df-dom 8888  df-sdom 8889  df-wdom 9473

This theorem is referenced by: (None)