Theorem xpcomen 3343

Description: Commutative law for equinumerosity of cross product. Proposition 4.22(d) of [Mendelson] p. 254.

Hypotheses

Ref	Expression
xpcomen.1	⊢ A ∈ V
xpcomen.2	⊢ B ∈ V

Assertion

Ref	Expression
xpcomen	⊢ (A × B) ≈ (B × A)

Proof of Theorem xpcomen

Step	Hyp	Ref	Expression
1		xpcomen.1	. . 3 ⊢ A ∈ V
2		xpcomen.2	. . 3 ⊢ B ∈ V
3	1, 2	xpex 2488	. 2 ⊢ (A × B) ∈ V
4		snex 1859	. . . . 5 ⊢ {x} ∈ V
5	4	cnvex 2670	. . . 4 ⊢ ◡{x} ∈ V
6	5	uniex 1947	. . 3 ⊢ ∪◡{x} ∈ V
7	6	a1i 7	. 2 ⊢ (x ∈ (A × B) → ∪◡{x} ∈ V)
8		snex 1859	. . . . 5 ⊢ {y} ∈ V
9	8	cnvex 2670	. . . 4 ⊢ ◡{y} ∈ V
10	9	uniex 1947	. . 3 ⊢ ∪◡{y} ∈ V
11	10	a1i 7	. 2 ⊢ (y ∈ (B × A) → ∪◡{y} ∈ V)
12		sneq 1816	. . . . . . . . . . . 12 ⊢ (x = ⟨z, w⟩ → {x} = {⟨z, w⟩})
13		cnveq 2513	. . . . . . . . . . . 12 ⊢ ({x} = {⟨z, w⟩} → ◡{x} = ◡{⟨z, w⟩})
14	12, 13	syl 12	. . . . . . . . . . 11 ⊢ (x = ⟨z, w⟩ → ◡{x} = ◡{⟨z, w⟩})
15		visset 1350	. . . . . . . . . . . 12 ⊢ z ∈ V
16		visset 1350	. . . . . . . . . . . 12 ⊢ w ∈ V
17	15, 16	cnvsn 2636	. . . . . . . . . . 11 ⊢ ◡{⟨z, w⟩} = {⟨w, z⟩}
18	14, 17	syl6eq 1140	. . . . . . . . . 10 ⊢ (x = ⟨z, w⟩ → ◡{x} = {⟨w, z⟩})
19	18	unieqd 1929	. . . . . . . . 9 ⊢ (x = ⟨z, w⟩ → ∪◡{x} = ∪{⟨w, z⟩})
20		opex 1893	. . . . . . . . . 10 ⊢ ⟨w, z⟩ ∈ V
21	20	unisn 1932	. . . . . . . . 9 ⊢ ∪{⟨w, z⟩} = ⟨w, z⟩
22	19, 21	syl6req 1141	. . . . . . . 8 ⊢ (x = ⟨z, w⟩ → ⟨w, z⟩ = ∪◡{x})
23		sneq 1816	. . . . . . . . . . . 12 ⊢ (y = ⟨w, z⟩ → {y} = {⟨w, z⟩})
24		cnveq 2513	. . . . . . . . . . . 12 ⊢ ({y} = {⟨w, z⟩} → ◡{y} = ◡{⟨w, z⟩})
25	23, 24	syl 12	. . . . . . . . . . 11 ⊢ (y = ⟨w, z⟩ → ◡{y} = ◡{⟨w, z⟩})
26	16, 15	cnvsn 2636	. . . . . . . . . . 11 ⊢ ◡{⟨w, z⟩} = {⟨z, w⟩}
27	25, 26	syl6eq 1140	. . . . . . . . . 10 ⊢ (y = ⟨w, z⟩ → ◡{y} = {⟨z, w⟩})
28	27	unieqd 1929	. . . . . . . . 9 ⊢ (y = ⟨w, z⟩ → ∪◡{y} = ∪{⟨z, w⟩})
29		opex 1893	. . . . . . . . . 10 ⊢ ⟨z, w⟩ ∈ V
30	29	unisn 1932	. . . . . . . . 9 ⊢ ∪{⟨z, w⟩} = ⟨z, w⟩
31	28, 30	syl6req 1141	. . . . . . . 8 ⊢ (y = ⟨w, z⟩ → ⟨z, w⟩ = ∪◡{y})
32	22, 31	cleq2tr 1148	. . . . . . 7 ⊢ ((x = ⟨z, w⟩ ∧ y = ∪◡{x}) ↔ (y = ⟨w, z⟩ ∧ x = ∪◡{y}))
33		ancom 333	. . . . . . 7 ⊢ ((z ∈ A ∧ w ∈ B) ↔ (w ∈ B ∧ z ∈ A))
34	32, 33	anbi12i 369	. . . . . 6 ⊢ (((x = ⟨z, w⟩ ∧ y = ∪◡{x}) ∧ (z ∈ A ∧ w ∈ B)) ↔ ((y = ⟨w, z⟩ ∧ x = ∪◡{y}) ∧ (w ∈ B ∧ z ∈ A)))
35		an23 371	. . . . . 6 ⊢ (((x = ⟨z, w⟩ ∧ (z ∈ A ∧ w ∈ B)) ∧ y = ∪◡{x}) ↔ ((x = ⟨z, w⟩ ∧ y = ∪◡{x}) ∧ (z ∈ A ∧ w ∈ B)))
36		an23 371	. . . . . 6 ⊢ (((y = ⟨w, z⟩ ∧ (w ∈ B ∧ z ∈ A)) ∧ x = ∪◡{y}) ↔ ((y = ⟨w, z⟩ ∧ x = ∪◡{y}) ∧ (w ∈ B ∧ z ∈ A)))
37	34, 35, 36	3bitr4 158	. . . . 5 ⊢ (((x = ⟨z, w⟩ ∧ (z ∈ A ∧ w ∈ B)) ∧ y = ∪◡{x}) ↔ ((y = ⟨w, z⟩ ∧ (w ∈ B ∧ z ∈ A)) ∧ x = ∪◡{y}))
38	37	bi2ex 734	. . . 4 ⊢ (∃z∃w((x = ⟨z, w⟩ ∧ (z ∈ A ∧ w ∈ B)) ∧ y = ∪◡{x}) ↔ ∃z∃w((y = ⟨w, z⟩ ∧ (w ∈ B ∧ z ∈ A)) ∧ x = ∪◡{y}))
39		19.41vv 964	. . . 4 ⊢ (∃z∃w((x = ⟨z, w⟩ ∧ (z ∈ A ∧ w ∈ B)) ∧ y = ∪◡{x}) ↔ (∃z∃w(x = ⟨z, w⟩ ∧ (z ∈ A ∧ w ∈ B)) ∧ y = ∪◡{x}))
40		19.41vv 964	. . . 4 ⊢ (∃z∃w((y = ⟨w, z⟩ ∧ (w ∈ B ∧ z ∈ A)) ∧ x = ∪◡{y}) ↔ (∃z∃w(y = ⟨w, z⟩ ∧ (w ∈ B ∧ z ∈ A)) ∧ x = ∪◡{y}))
41	38, 39, 40	3bitr3 156	. . 3 ⊢ ((∃z∃w(x = ⟨z, w⟩ ∧ (z ∈ A ∧ w ∈ B)) ∧ y = ∪◡{x}) ↔ (∃z∃w(y = ⟨w, z⟩ ∧ (w ∈ B ∧ z ∈ A)) ∧ x = ∪◡{y}))
42		elxp 2442	. . . 4 ⊢ (x ∈ (A × B) ↔ ∃z∃w(x = ⟨z, w⟩ ∧ (z ∈ A ∧ w ∈ B)))
43	42	anbi1i 368	. . 3 ⊢ ((x ∈ (A × B) ∧ y = ∪◡{x}) ↔ (∃z∃w(x = ⟨z, w⟩ ∧ (z ∈ A ∧ w ∈ B)) ∧ y = ∪◡{x}))
44		elxp 2442	. . . . 5 ⊢ (y ∈ (B × A) ↔ ∃w∃z(y = ⟨w, z⟩ ∧ (w ∈ B ∧ z ∈ A)))
45		excom 728	. . . . 5 ⊢ (∃w∃z(y = ⟨w, z⟩ ∧ (w ∈ B ∧ z ∈ A)) ↔ ∃z∃w(y = ⟨w, z⟩ ∧ (w ∈ B ∧ z ∈ A)))
46	44, 45	bitr 151	. . . 4 ⊢ (y ∈ (B × A) ↔ ∃z∃w(y = ⟨w, z⟩ ∧ (w ∈ B ∧ z ∈ A)))
47	46	anbi1i 368	. . 3 ⊢ ((y ∈ (B × A) ∧ x = ∪◡{y}) ↔ (∃z∃w(y = ⟨w, z⟩ ∧ (w ∈ B ∧ z ∈ A)) ∧ x = ∪◡{y}))
48	41, 43, 47	3bitr4 158	. 2 ⊢ ((x ∈ (A × B) ∧ y = ∪◡{x}) ↔ (y ∈ (B × A) ∧ x = ∪◡{y}))
49	3, 7, 11, 48	en2 3305	1 ⊢ (A × B) ≈ (B × A)

Syntax hints:   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {csn 1808  ⟨cop 1810  ∪cuni 1919   class class class wbr 2054   × cxp 2408  ^◡ccnv 2409   ≈ cen 3271

This theorem is referenced by:  xpdom1 3346  xpen 3383  cdaassen 3725  infmap2 4953

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-en 3274

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