Theorem xpsnen 8595

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		snex 5323	. . 3 ⊢ {𝐵} ∈ V
3	1, 2	xpex 7470	. 2 ⊢ (𝐴 × {𝐵}) ∈ V
4		elxp 5572	. . 3 ⊢ (𝑦 ∈ (𝐴 × {𝐵}) ↔ ∃𝑥∃𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})))
5		inteq 4871	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑥, 𝑧⟩ → ∩ 𝑦 = ∩ ⟨𝑥, 𝑧⟩)
6	5	inteqd 4873	. . . . . . 7 ⊢ (𝑦 = ⟨𝑥, 𝑧⟩ → ∩ ∩ 𝑦 = ∩ ∩ ⟨𝑥, 𝑧⟩)
7		vex 3497	. . . . . . . 8 ⊢ 𝑥 ∈ V
8		vex 3497	. . . . . . . 8 ⊢ 𝑧 ∈ V
9	7, 8	op1stb 5355	. . . . . . 7 ⊢ ∩ ∩ ⟨𝑥, 𝑧⟩ = 𝑥
10	6, 9	syl6eq 2872	. . . . . 6 ⊢ (𝑦 = ⟨𝑥, 𝑧⟩ → ∩ ∩ 𝑦 = 𝑥)
11	10, 7	eqeltrdi 2921	. . . . 5 ⊢ (𝑦 = ⟨𝑥, 𝑧⟩ → ∩ ∩ 𝑦 ∈ V)
12	11	adantr 483	. . . 4 ⊢ ((𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})) → ∩ ∩ 𝑦 ∈ V)
13	12	exlimivv 1929	. . 3 ⊢ (∃𝑥∃𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})) → ∩ ∩ 𝑦 ∈ V)
14	4, 13	sylbi 219	. 2 ⊢ (𝑦 ∈ (𝐴 × {𝐵}) → ∩ ∩ 𝑦 ∈ V)
15		opex 5348	. . 3 ⊢ ⟨𝑥, 𝐵⟩ ∈ V
16	15	a1i 11	. 2 ⊢ (𝑥 ∈ 𝐴 → ⟨𝑥, 𝐵⟩ ∈ V)
17		eqvisset 3511	. . . . 5 ⊢ (𝑥 = ∩ ∩ 𝑦 → ∩ ∩ 𝑦 ∈ V)
18		ancom 463	. . . . . . . . . . 11 ⊢ (((𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ {𝐵}) ↔ (𝑧 ∈ {𝐵} ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴)))
19		anass 471	. . . . . . . . . . 11 ⊢ (((𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ {𝐵}) ↔ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})))
20		velsn 4576	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵)
21	20	anbi1i 625	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ {𝐵} ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴)) ↔ (𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴)))
22	18, 19, 21	3bitr3i 303	. . . . . . . . . 10 ⊢ ((𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})) ↔ (𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴)))
23	22	exbii 1844	. . . . . . . . 9 ⊢ (∃𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})) ↔ ∃𝑧(𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴)))
24		xpsnen.2	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
25		opeq2 4797	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐵 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝐵⟩)
26	25	eqeq2d 2832	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → (𝑦 = ⟨𝑥, 𝑧⟩ ↔ 𝑦 = ⟨𝑥, 𝐵⟩))
27	26	anbi1d 631	. . . . . . . . . 10 ⊢ (𝑧 = 𝐵 → ((𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴) ↔ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴)))
28	24, 27	ceqsexv 3541	. . . . . . . . 9 ⊢ (∃𝑧(𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴)) ↔ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴))
29		inteq 4871	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ⟨𝑥, 𝐵⟩ → ∩ 𝑦 = ∩ ⟨𝑥, 𝐵⟩)
30	29	inteqd 4873	. . . . . . . . . . . . 13 ⊢ (𝑦 = ⟨𝑥, 𝐵⟩ → ∩ ∩ 𝑦 = ∩ ∩ ⟨𝑥, 𝐵⟩)
31	7, 24	op1stb 5355	. . . . . . . . . . . . 13 ⊢ ∩ ∩ ⟨𝑥, 𝐵⟩ = 𝑥
32	30, 31	syl6req 2873	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑥, 𝐵⟩ → 𝑥 = ∩ ∩ 𝑦)
33	32	pm4.71ri 563	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑥, 𝐵⟩ ↔ (𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑥, 𝐵⟩))
34	33	anbi1i 625	. . . . . . . . . 10 ⊢ ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) ↔ ((𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑥, 𝐵⟩) ∧ 𝑥 ∈ 𝐴))
35		anass 471	. . . . . . . . . 10 ⊢ (((𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑥, 𝐵⟩) ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 = ∩ ∩ 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴)))
36	34, 35	bitri 277	. . . . . . . . 9 ⊢ ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 = ∩ ∩ 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴)))
37	23, 28, 36	3bitri 299	. . . . . . . 8 ⊢ (∃𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})) ↔ (𝑥 = ∩ ∩ 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴)))
38	37	exbii 1844	. . . . . . 7 ⊢ (∃𝑥∃𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})) ↔ ∃𝑥(𝑥 = ∩ ∩ 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴)))
39	4, 38	bitri 277	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 × {𝐵}) ↔ ∃𝑥(𝑥 = ∩ ∩ 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴)))
40		opeq1 4796	. . . . . . . . 9 ⊢ (𝑥 = ∩ ∩ 𝑦 → ⟨𝑥, 𝐵⟩ = ⟨∩ ∩ 𝑦, 𝐵⟩)
41	40	eqeq2d 2832	. . . . . . . 8 ⊢ (𝑥 = ∩ ∩ 𝑦 → (𝑦 = ⟨𝑥, 𝐵⟩ ↔ 𝑦 = ⟨∩ ∩ 𝑦, 𝐵⟩))
42		eleq1 2900	. . . . . . . 8 ⊢ (𝑥 = ∩ ∩ 𝑦 → (𝑥 ∈ 𝐴 ↔ ∩ ∩ 𝑦 ∈ 𝐴))
43	41, 42	anbi12d 632	. . . . . . 7 ⊢ (𝑥 = ∩ ∩ 𝑦 → ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) ↔ (𝑦 = ⟨∩ ∩ 𝑦, 𝐵⟩ ∧ ∩ ∩ 𝑦 ∈ 𝐴)))
44	43	ceqsexgv 3646	. . . . . 6 ⊢ (∩ ∩ 𝑦 ∈ V → (∃𝑥(𝑥 = ∩ ∩ 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴)) ↔ (𝑦 = ⟨∩ ∩ 𝑦, 𝐵⟩ ∧ ∩ ∩ 𝑦 ∈ 𝐴)))
45	39, 44	syl5bb 285	. . . . 5 ⊢ (∩ ∩ 𝑦 ∈ V → (𝑦 ∈ (𝐴 × {𝐵}) ↔ (𝑦 = ⟨∩ ∩ 𝑦, 𝐵⟩ ∧ ∩ ∩ 𝑦 ∈ 𝐴)))
46	17, 45	syl 17	. . . 4 ⊢ (𝑥 = ∩ ∩ 𝑦 → (𝑦 ∈ (𝐴 × {𝐵}) ↔ (𝑦 = ⟨∩ ∩ 𝑦, 𝐵⟩ ∧ ∩ ∩ 𝑦 ∈ 𝐴)))
47	46	pm5.32ri 578	. . 3 ⊢ ((𝑦 ∈ (𝐴 × {𝐵}) ∧ 𝑥 = ∩ ∩ 𝑦) ↔ ((𝑦 = ⟨∩ ∩ 𝑦, 𝐵⟩ ∧ ∩ ∩ 𝑦 ∈ 𝐴) ∧ 𝑥 = ∩ ∩ 𝑦))
48	32	adantr 483	. . . . 5 ⊢ ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) → 𝑥 = ∩ ∩ 𝑦)
49	48	pm4.71i 562	. . . 4 ⊢ ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) ↔ ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = ∩ ∩ 𝑦))
50	43	pm5.32ri 578	. . . 4 ⊢ (((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = ∩ ∩ 𝑦) ↔ ((𝑦 = ⟨∩ ∩ 𝑦, 𝐵⟩ ∧ ∩ ∩ 𝑦 ∈ 𝐴) ∧ 𝑥 = ∩ ∩ 𝑦))
51	49, 50	bitr2i 278	. . 3 ⊢ (((𝑦 = ⟨∩ ∩ 𝑦, 𝐵⟩ ∧ ∩ ∩ 𝑦 ∈ 𝐴) ∧ 𝑥 = ∩ ∩ 𝑦) ↔ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴))
52		ancom 463	. . 3 ⊢ ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = ⟨𝑥, 𝐵⟩))
53	47, 51, 52	3bitri 299	. 2 ⊢ ((𝑦 ∈ (𝐴 × {𝐵}) ∧ 𝑥 = ∩ ∩ 𝑦) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = ⟨𝑥, 𝐵⟩))
54	3, 1, 14, 16, 53	en2i 8541	1 ⊢ (𝐴 × {𝐵}) ≈ 𝐴

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  Vcvv 3494  {csn 4560  ⟨cop 4566  ∩ cint 4868   class class class wbr 5058   × cxp 5547   ≈ cen 8500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-int 4869  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-en 8504

This theorem is referenced by:  xpsneng  8596  endisj  8598  infxpenlem  9433  hashxplem  13788  rexpen  15575  heiborlem3  35085