Theorem xpsnen 6715

Description: A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)

Hypotheses

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

Assertion

Ref	Expression
xpsnen	⊢ (𝐴 × {𝐵}) ≈ 𝐴

Proof of Theorem xpsnen

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpsnen.1	. . 3 ⊢ 𝐴 ∈ V
2		xpsnen.2	. . . 4 ⊢ 𝐵 ∈ V
3	2	snex 4109	. . 3 ⊢ {𝐵} ∈ V
4	1, 3	xpex 4654	. 2 ⊢ (𝐴 × {𝐵}) ∈ V
5		elxp 4556	. . 3 ⊢ (𝑦 ∈ (𝐴 × {𝐵}) ↔ ∃𝑥∃𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})))
6		inteq 3774	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑥, 𝑧⟩ → ∩ 𝑦 = ∩ ⟨𝑥, 𝑧⟩)
7	6	inteqd 3776	. . . . . . 7 ⊢ (𝑦 = ⟨𝑥, 𝑧⟩ → ∩ ∩ 𝑦 = ∩ ∩ ⟨𝑥, 𝑧⟩)
8		vex 2689	. . . . . . . 8 ⊢ 𝑥 ∈ V
9		vex 2689	. . . . . . . 8 ⊢ 𝑧 ∈ V
10	8, 9	op1stb 4399	. . . . . . 7 ⊢ ∩ ∩ ⟨𝑥, 𝑧⟩ = 𝑥
11	7, 10	syl6eq 2188	. . . . . 6 ⊢ (𝑦 = ⟨𝑥, 𝑧⟩ → ∩ ∩ 𝑦 = 𝑥)
12	11, 8	eqeltrdi 2230	. . . . 5 ⊢ (𝑦 = ⟨𝑥, 𝑧⟩ → ∩ ∩ 𝑦 ∈ V)
13	12	adantr 274	. . . 4 ⊢ ((𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})) → ∩ ∩ 𝑦 ∈ V)
14	13	exlimivv 1868	. . 3 ⊢ (∃𝑥∃𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})) → ∩ ∩ 𝑦 ∈ V)
15	5, 14	sylbi 120	. 2 ⊢ (𝑦 ∈ (𝐴 × {𝐵}) → ∩ ∩ 𝑦 ∈ V)
16	8, 2	opex 4151	. . 3 ⊢ ⟨𝑥, 𝐵⟩ ∈ V
17	16	a1i 9	. 2 ⊢ (𝑥 ∈ 𝐴 → ⟨𝑥, 𝐵⟩ ∈ V)
18		eqvisset 2696	. . . . 5 ⊢ (𝑥 = ∩ ∩ 𝑦 → ∩ ∩ 𝑦 ∈ V)
19		ancom 264	. . . . . . . . . . 11 ⊢ (((𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ {𝐵}) ↔ (𝑧 ∈ {𝐵} ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴)))
20		anass 398	. . . . . . . . . . 11 ⊢ (((𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ {𝐵}) ↔ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})))
21		velsn 3544	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵)
22	21	anbi1i 453	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ {𝐵} ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴)) ↔ (𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴)))
23	19, 20, 22	3bitr3i 209	. . . . . . . . . 10 ⊢ ((𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})) ↔ (𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴)))
24	23	exbii 1584	. . . . . . . . 9 ⊢ (∃𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})) ↔ ∃𝑧(𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴)))
25		opeq2 3706	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐵 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝐵⟩)
26	25	eqeq2d 2151	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → (𝑦 = ⟨𝑥, 𝑧⟩ ↔ 𝑦 = ⟨𝑥, 𝐵⟩))
27	26	anbi1d 460	. . . . . . . . . 10 ⊢ (𝑧 = 𝐵 → ((𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴) ↔ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴)))
28	2, 27	ceqsexv 2725	. . . . . . . . 9 ⊢ (∃𝑧(𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴)) ↔ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴))
29		inteq 3774	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ⟨𝑥, 𝐵⟩ → ∩ 𝑦 = ∩ ⟨𝑥, 𝐵⟩)
30	29	inteqd 3776	. . . . . . . . . . . . 13 ⊢ (𝑦 = ⟨𝑥, 𝐵⟩ → ∩ ∩ 𝑦 = ∩ ∩ ⟨𝑥, 𝐵⟩)
31	8, 2	op1stb 4399	. . . . . . . . . . . . 13 ⊢ ∩ ∩ ⟨𝑥, 𝐵⟩ = 𝑥
32	30, 31	syl6req 2189	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑥, 𝐵⟩ → 𝑥 = ∩ ∩ 𝑦)
33	32	pm4.71ri 389	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑥, 𝐵⟩ ↔ (𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑥, 𝐵⟩))
34	33	anbi1i 453	. . . . . . . . . 10 ⊢ ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) ↔ ((𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑥, 𝐵⟩) ∧ 𝑥 ∈ 𝐴))
35		anass 398	. . . . . . . . . 10 ⊢ (((𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑥, 𝐵⟩) ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 = ∩ ∩ 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴)))
36	34, 35	bitri 183	. . . . . . . . 9 ⊢ ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 = ∩ ∩ 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴)))
37	24, 28, 36	3bitri 205	. . . . . . . 8 ⊢ (∃𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})) ↔ (𝑥 = ∩ ∩ 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴)))
38	37	exbii 1584	. . . . . . 7 ⊢ (∃𝑥∃𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})) ↔ ∃𝑥(𝑥 = ∩ ∩ 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴)))
39	5, 38	bitri 183	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 × {𝐵}) ↔ ∃𝑥(𝑥 = ∩ ∩ 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴)))
40		opeq1 3705	. . . . . . . . 9 ⊢ (𝑥 = ∩ ∩ 𝑦 → ⟨𝑥, 𝐵⟩ = ⟨∩ ∩ 𝑦, 𝐵⟩)
41	40	eqeq2d 2151	. . . . . . . 8 ⊢ (𝑥 = ∩ ∩ 𝑦 → (𝑦 = ⟨𝑥, 𝐵⟩ ↔ 𝑦 = ⟨∩ ∩ 𝑦, 𝐵⟩))
42		eleq1 2202	. . . . . . . 8 ⊢ (𝑥 = ∩ ∩ 𝑦 → (𝑥 ∈ 𝐴 ↔ ∩ ∩ 𝑦 ∈ 𝐴))
43	41, 42	anbi12d 464	. . . . . . 7 ⊢ (𝑥 = ∩ ∩ 𝑦 → ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) ↔ (𝑦 = ⟨∩ ∩ 𝑦, 𝐵⟩ ∧ ∩ ∩ 𝑦 ∈ 𝐴)))
44	43	ceqsexgv 2814	. . . . . 6 ⊢ (∩ ∩ 𝑦 ∈ V → (∃𝑥(𝑥 = ∩ ∩ 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴)) ↔ (𝑦 = ⟨∩ ∩ 𝑦, 𝐵⟩ ∧ ∩ ∩ 𝑦 ∈ 𝐴)))
45	39, 44	syl5bb 191	. . . . 5 ⊢ (∩ ∩ 𝑦 ∈ V → (𝑦 ∈ (𝐴 × {𝐵}) ↔ (𝑦 = ⟨∩ ∩ 𝑦, 𝐵⟩ ∧ ∩ ∩ 𝑦 ∈ 𝐴)))
46	18, 45	syl 14	. . . 4 ⊢ (𝑥 = ∩ ∩ 𝑦 → (𝑦 ∈ (𝐴 × {𝐵}) ↔ (𝑦 = ⟨∩ ∩ 𝑦, 𝐵⟩ ∧ ∩ ∩ 𝑦 ∈ 𝐴)))
47	46	pm5.32ri 450	. . 3 ⊢ ((𝑦 ∈ (𝐴 × {𝐵}) ∧ 𝑥 = ∩ ∩ 𝑦) ↔ ((𝑦 = ⟨∩ ∩ 𝑦, 𝐵⟩ ∧ ∩ ∩ 𝑦 ∈ 𝐴) ∧ 𝑥 = ∩ ∩ 𝑦))
48	32	adantr 274	. . . . 5 ⊢ ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) → 𝑥 = ∩ ∩ 𝑦)
49	48	pm4.71i 388	. . . 4 ⊢ ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) ↔ ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = ∩ ∩ 𝑦))
50	43	pm5.32ri 450	. . . 4 ⊢ (((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = ∩ ∩ 𝑦) ↔ ((𝑦 = ⟨∩ ∩ 𝑦, 𝐵⟩ ∧ ∩ ∩ 𝑦 ∈ 𝐴) ∧ 𝑥 = ∩ ∩ 𝑦))
51	49, 50	bitr2i 184	. . 3 ⊢ (((𝑦 = ⟨∩ ∩ 𝑦, 𝐵⟩ ∧ ∩ ∩ 𝑦 ∈ 𝐴) ∧ 𝑥 = ∩ ∩ 𝑦) ↔ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴))
52		ancom 264	. . 3 ⊢ ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = ⟨𝑥, 𝐵⟩))
53	47, 51, 52	3bitri 205	. 2 ⊢ ((𝑦 ∈ (𝐴 × {𝐵}) ∧ 𝑥 = ∩ ∩ 𝑦) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = ⟨𝑥, 𝐵⟩))
54	4, 1, 15, 17, 53	en2i 6664	1 ⊢ (𝐴 × {𝐵}) ≈ 𝐴

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480  Vcvv 2686  {csn 3527  ⟨cop 3530  ∩ cint 3771   class class class wbr 3929   × cxp 4537   ≈ cen 6632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-en 6635

This theorem is referenced by:  xpsneng  6716  endisj  6718