Theorem xpcomen 4445

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 3266	. 2 ⊢ (A × B) ∈ V
4		snex 2756	. . . . 5 ⊢ {x} ∈ V
5	4	cnvex 3526	. . . 4 ⊢ ◡{x} ∈ V
6	5	uniex 2876	. . 3 ⊢ ∪◡{x} ∈ V
7	6	a1i 8	. 2 ⊢ (x ∈ (A × B) → ∪◡{x} ∈ V)
8		snex 2756	. . . . 5 ⊢ {y} ∈ V
9	8	cnvex 3526	. . . 4 ⊢ ◡{y} ∈ V
10	9	uniex 2876	. . 3 ⊢ ∪◡{y} ∈ V
11	10	a1i 8	. 2 ⊢ (y ∈ (B × A) → ∪◡{y} ∈ V)
12		sneq 2421	. . . . . . . . . . . 12 ⊢ (x = ⟨z, w⟩ → {x} = {⟨z, w⟩})
13		cnveq 3298	. . . . . . . . . . . 12 ⊢ ({x} = {⟨z, w⟩} → ◡{x} = ◡{⟨z, w⟩})
14	12, 13	syl 10	. . . . . . . . . . 11 ⊢ (x = ⟨z, w⟩ → ◡{x} = ◡{⟨z, w⟩})
15		visset 1816	. . . . . . . . . . . 12 ⊢ z ∈ V
16		visset 1816	. . . . . . . . . . . 12 ⊢ w ∈ V
17	15, 16	cnvsn 3455	. . . . . . . . . . 11 ⊢ ◡{⟨z, w⟩} = {⟨w, z⟩}
18	14, 17	syl6eq 1526	. . . . . . . . . 10 ⊢ (x = ⟨z, w⟩ → ◡{x} = {⟨w, z⟩})
19	18	unieqd 2516	. . . . . . . . 9 ⊢ (x = ⟨z, w⟩ → ∪◡{x} = ∪{⟨w, z⟩})
20		opex 2788	. . . . . . . . . 10 ⊢ ⟨w, z⟩ ∈ V
21	20	unisn 2521	. . . . . . . . 9 ⊢ ∪{⟨w, z⟩} = ⟨w, z⟩
22	19, 21	syl6req 1527	. . . . . . . 8 ⊢ (x = ⟨z, w⟩ → ⟨w, z⟩ = ∪◡{x})
23		sneq 2421	. . . . . . . . . . . 12 ⊢ (y = ⟨w, z⟩ → {y} = {⟨w, z⟩})
24		cnveq 3298	. . . . . . . . . . . 12 ⊢ ({y} = {⟨w, z⟩} → ◡{y} = ◡{⟨w, z⟩})
25	23, 24	syl 10	. . . . . . . . . . 11 ⊢ (y = ⟨w, z⟩ → ◡{y} = ◡{⟨w, z⟩})
26	16, 15	cnvsn 3455	. . . . . . . . . . 11 ⊢ ◡{⟨w, z⟩} = {⟨z, w⟩}
27	25, 26	syl6eq 1526	. . . . . . . . . 10 ⊢ (y = ⟨w, z⟩ → ◡{y} = {⟨z, w⟩})
28	27	unieqd 2516	. . . . . . . . 9 ⊢ (y = ⟨w, z⟩ → ∪◡{y} = ∪{⟨z, w⟩})
29		opex 2788	. . . . . . . . . 10 ⊢ ⟨z, w⟩ ∈ V
30	29	unisn 2521	. . . . . . . . 9 ⊢ ∪{⟨z, w⟩} = ⟨z, w⟩
31	28, 30	syl6req 1527	. . . . . . . 8 ⊢ (y = ⟨w, z⟩ → ⟨z, w⟩ = ∪◡{y})
32	22, 31	eq2tr 1536	. . . . . . 7 ⊢ ((x = ⟨z, w⟩ ⋀ y = ∪◡{x}) ↔ (y = ⟨w, z⟩ ⋀ x = ∪◡{y}))
33		ancom 437	. . . . . . 7 ⊢ ((z ∈ A ⋀ w ∈ B) ↔ (w ∈ B ⋀ z ∈ A))
34	32, 33	anbi12i 484	. . . . . 6 ⊢ (((x = ⟨z, w⟩ ⋀ y = ∪◡{x}) ⋀ (z ∈ A ⋀ w ∈ B)) ↔ ((y = ⟨w, z⟩ ⋀ x = ∪◡{y}) ⋀ (w ∈ B ⋀ z ∈ A)))
35		an23 487	. . . . . 6 ⊢ (((x = ⟨z, w⟩ ⋀ (z ∈ A ⋀ w ∈ B)) ⋀ y = ∪◡{x}) ↔ ((x = ⟨z, w⟩ ⋀ y = ∪◡{x}) ⋀ (z ∈ A ⋀ w ∈ B)))
36		an23 487	. . . . . 6 ⊢ (((y = ⟨w, z⟩ ⋀ (w ∈ B ⋀ z ∈ A)) ⋀ x = ∪◡{y}) ↔ ((y = ⟨w, z⟩ ⋀ x = ∪◡{y}) ⋀ (w ∈ B ⋀ z ∈ A)))
37	34, 35, 36	3bitr4 183	. . . . 5 ⊢ (((x = ⟨z, w⟩ ⋀ (z ∈ A ⋀ w ∈ B)) ⋀ y = ∪◡{x}) ↔ ((y = ⟨w, z⟩ ⋀ (w ∈ B ⋀ z ∈ A)) ⋀ x = ∪◡{y}))
38	37	2exbii 1054	. . . 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 1308	. . . 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 1308	. . . 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 181	. . 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 3208	. . . 4 ⊢ (x ∈ (A × B) ↔ ∃z∃w(x = ⟨z, w⟩ ⋀ (z ∈ A ⋀ w ∈ B)))
43	42	anbi1i 483	. . 3 ⊢ ((x ∈ (A × B) ⋀ y = ∪◡{x}) ↔ (∃z∃w(x = ⟨z, w⟩ ⋀ (z ∈ A ⋀ w ∈ B)) ⋀ y = ∪◡{x}))
44		elxp 3208	. . . . 5 ⊢ (y ∈ (B × A) ↔ ∃w∃z(y = ⟨w, z⟩ ⋀ (w ∈ B ⋀ z ∈ A)))
45		excom 1048	. . . . 5 ⊢ (∃w∃z(y = ⟨w, z⟩ ⋀ (w ∈ B ⋀ z ∈ A)) ↔ ∃z∃w(y = ⟨w, z⟩ ⋀ (w ∈ B ⋀ z ∈ A)))
46	44, 45	bitr 173	. . . 4 ⊢ (y ∈ (B × A) ↔ ∃z∃w(y = ⟨w, z⟩ ⋀ (w ∈ B ⋀ z ∈ A)))
47	46	anbi1i 483	. . 3 ⊢ ((y ∈ (B × A) ⋀ x = ∪◡{y}) ↔ (∃z∃w(y = ⟨w, z⟩ ⋀ (w ∈ B ⋀ z ∈ A)) ⋀ x = ∪◡{y}))
48	41, 43, 47	3bitr4 183	. 2 ⊢ ((x ∈ (A × B) ⋀ y = ∪◡{x}) ↔ (y ∈ (B × A) ⋀ x = ∪◡{y}))
49	3, 7, 11, 48	en2 4408	1 ⊢ (A × B) ≈ (B × A)

Syntax hints:   ⋀ wa 223   = wceq 958   ∈ wcel 960  ∃wex 982  Vcvv 1814  {csn 2413  ⟨cop 2415  ∪cuni 2507   class class class wbr 2624   × cxp 3174  ^◡ccnv 3175   ≈ cen 4370

This theorem is referenced by:  xpcomeng 4446  xpdom1 4449  xpen 4494  cdaassen 4942  infxp 7573  iunctb 7576  infmap2 7583

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-en 4374

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