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	⊢ (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		xpexg 7106	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐵 × 𝐶) ∈ V)
5	2, 3, 4	syl2anc 565	. 2 ⊢ (𝜑 → (𝐵 × 𝐶) ∈ V)
6		wdom2d2.o	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = 𝑋)
7		nfcsb1v 3696	. . . . 5 ⊢ Ⅎ𝑦⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋
8	7	nfeq2 2928	. . . 4 ⊢ Ⅎ𝑦 𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋
9		nfcv 2912	. . . . . 6 ⊢ Ⅎ𝑧(1st ‘𝑤)
10		nfcsb1v 3696	. . . . . 6 ⊢ Ⅎ𝑧⦋(2nd ‘𝑤) / 𝑧⦌𝑋
11	9, 10	nfcsb 3698	. . . . 5 ⊢ Ⅎ𝑧⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋
12	11	nfeq2 2928	. . . 4 ⊢ Ⅎ𝑧 𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋
13		nfv 1994	. . . 4 ⊢ Ⅎ𝑤 𝑥 = 𝑋
14		csbopeq1a 7369	. . . . 5 ⊢ (𝑤 = ⟨𝑦, 𝑧⟩ → ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 = 𝑋)
15	14	eqeq2d 2780	. . . 4 ⊢ (𝑤 = ⟨𝑦, 𝑧⟩ → (𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 ↔ 𝑥 = 𝑋))
16	8, 12, 13, 15	rexxpf 5408	. . 3 ⊢ (∃𝑤 ∈ (𝐵 × 𝐶)𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋 ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = 𝑋)
17	6, 16	sylibr 224	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑤 ∈ (𝐵 × 𝐶)𝑥 = ⦋(1st ‘𝑤) / 𝑦⦌⦋(2nd ‘𝑤) / 𝑧⦌𝑋)
18	1, 5, 17	wdom2d 8640	1 ⊢ (𝜑 → 𝐴 ≼* (𝐵 × 𝐶))

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  1^st c1st 7312  2^nd 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)