Theorem wdom2d2 44021

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 7763	. 2 ⊢ (𝜑 → (𝐵 × 𝐶) ∈ V)
5		wdom2d2.o	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = 𝑋)
6		nfcsb1v 3871	. . . . 5 ⊢ Ⅎ𝑦⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋
7	6	nfeq2 2940	. . . 4 ⊢ Ⅎ𝑦 𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋
8		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑧(1st ‘𝑤)
9		nfcsb1v 3871	. . . . . 6 ⊢ Ⅎ𝑧⦋(2nd ‘𝑤) / 𝑧⦌𝑋
10	8, 9	nfcsbw 3873	. . . . 5 ⊢ Ⅎ𝑧⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋
11	10	nfeq2 2940	. . . 4 ⊢ Ⅎ𝑧 𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋
12		nfv 1947	. . . 4 ⊢ Ⅎ𝑤 𝑥 = 𝑋
13		csbopeq1a 8059	. . . . 5 ⊢ (𝑤 = ⟨𝑦, 𝑧⟩ → ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 = 𝑋)
14	13	eqeq2d 2772	. . . 4 ⊢ (𝑤 = ⟨𝑦, 𝑧⟩ → (𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 ↔ 𝑥 = 𝑋))
15	7, 11, 12, 14	rexxpf 5825	. . 3 ⊢ (∃𝑤 ∈ (𝐵 × 𝐶)𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = 𝑋)
16	5, 15	sylibr 237	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑤 ∈ (𝐵 × 𝐶)𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋)
17	1, 4, 16	wdom2d 9567	1 ⊢ (𝜑 → 𝐴 ≼* (𝐵 × 𝐶))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3087  Vcvv 3451  ⦋csb 3847  ⟨cop 4590   class class class wbr 5103   × cxp 5649  ‘cfv 6537  1^st c1st 7997  2^nd c2nd 7998   ≼^* cwdom 9551

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-1st 7999  df-2nd 8000  df-en 8967  df-dom 8968  df-sdom 8969  df-wdom 9552

This theorem is used by: (None)