Theorem xpsnen 9095

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 5436	. . 3 ⊢ {𝐵} ∈ V
3	1, 2	xpex 7773	. 2 ⊢ (𝐴 × {𝐵}) ∈ V
4		elxp 5708	. . 3 ⊢ (𝑦 ∈ (𝐴 × {𝐵}) ↔ ∃𝑥∃𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})))
5		inteq 4949	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑥, 𝑧⟩ → ∩ 𝑦 = ∩ ⟨𝑥, 𝑧⟩)
6	5	inteqd 4951	. . . . . . 7 ⊢ (𝑦 = ⟨𝑥, 𝑧⟩ → ∩ ∩ 𝑦 = ∩ ∩ ⟨𝑥, 𝑧⟩)
7		vex 3484	. . . . . . . 8 ⊢ 𝑥 ∈ V
8		vex 3484	. . . . . . . 8 ⊢ 𝑧 ∈ V
9	7, 8	op1stb 5476	. . . . . . 7 ⊢ ∩ ∩ ⟨𝑥, 𝑧⟩ = 𝑥
10	6, 9	eqtrdi 2793	. . . . . 6 ⊢ (𝑦 = ⟨𝑥, 𝑧⟩ → ∩ ∩ 𝑦 = 𝑥)
11	10, 7	eqeltrdi 2849	. . . . 5 ⊢ (𝑦 = ⟨𝑥, 𝑧⟩ → ∩ ∩ 𝑦 ∈ V)
12	11	adantr 480	. . . 4 ⊢ ((𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})) → ∩ ∩ 𝑦 ∈ V)
13	12	exlimivv 1932	. . 3 ⊢ (∃𝑥∃𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})) → ∩ ∩ 𝑦 ∈ V)
14	4, 13	sylbi 217	. 2 ⊢ (𝑦 ∈ (𝐴 × {𝐵}) → ∩ ∩ 𝑦 ∈ V)
15		opex 5469	. . 3 ⊢ ⟨𝑥, 𝐵⟩ ∈ V
16	15	a1i 11	. 2 ⊢ (𝑥 ∈ 𝐴 → ⟨𝑥, 𝐵⟩ ∈ V)
17		eqvisset 3500	. . . . 5 ⊢ (𝑥 = ∩ ∩ 𝑦 → ∩ ∩ 𝑦 ∈ V)
18		ancom 460	. . . . . . . . . . 11 ⊢ (((𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ {𝐵}) ↔ (𝑧 ∈ {𝐵} ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴)))
19		anass 468	. . . . . . . . . . 11 ⊢ (((𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ {𝐵}) ↔ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})))
20		velsn 4642	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵)
21	20	anbi1i 624	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ {𝐵} ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴)) ↔ (𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴)))
22	18, 19, 21	3bitr3i 301	. . . . . . . . . 10 ⊢ ((𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})) ↔ (𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴)))
23	22	exbii 1848	. . . . . . . . 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 4874	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐵 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝐵⟩)
26	25	eqeq2d 2748	. . . . . . . . . . 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 3532	. . . . . . . . 9 ⊢ (∃𝑧(𝑧 = 𝐵 ∧ (𝑦 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 ∈ 𝐴)) ↔ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴))
29		inteq 4949	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ⟨𝑥, 𝐵⟩ → ∩ 𝑦 = ∩ ⟨𝑥, 𝐵⟩)
30	29	inteqd 4951	. . . . . . . . . . . . 13 ⊢ (𝑦 = ⟨𝑥, 𝐵⟩ → ∩ ∩ 𝑦 = ∩ ∩ ⟨𝑥, 𝐵⟩)
31	7, 24	op1stb 5476	. . . . . . . . . . . . 13 ⊢ ∩ ∩ ⟨𝑥, 𝐵⟩ = 𝑥
32	30, 31	eqtr2di 2794	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑥, 𝐵⟩ → 𝑥 = ∩ ∩ 𝑦)
33	32	pm4.71ri 560	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑥, 𝐵⟩ ↔ (𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑥, 𝐵⟩))
34	33	anbi1i 624	. . . . . . . . . 10 ⊢ ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) ↔ ((𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑥, 𝐵⟩) ∧ 𝑥 ∈ 𝐴))
35		anass 468	. . . . . . . . . 10 ⊢ (((𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑥, 𝐵⟩) ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 = ∩ ∩ 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴)))
36	34, 35	bitri 275	. . . . . . . . 9 ⊢ ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 = ∩ ∩ 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴)))
37	23, 28, 36	3bitri 297	. . . . . . . 8 ⊢ (∃𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})) ↔ (𝑥 = ∩ ∩ 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴)))
38	37	exbii 1848	. . . . . . 7 ⊢ (∃𝑥∃𝑧(𝑦 = ⟨𝑥, 𝑧⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ {𝐵})) ↔ ∃𝑥(𝑥 = ∩ ∩ 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴)))
39	4, 38	bitri 275	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 × {𝐵}) ↔ ∃𝑥(𝑥 = ∩ ∩ 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴)))
40		opeq1 4873	. . . . . . . . 9 ⊢ (𝑥 = ∩ ∩ 𝑦 → ⟨𝑥, 𝐵⟩ = ⟨∩ ∩ 𝑦, 𝐵⟩)
41	40	eqeq2d 2748	. . . . . . . 8 ⊢ (𝑥 = ∩ ∩ 𝑦 → (𝑦 = ⟨𝑥, 𝐵⟩ ↔ 𝑦 = ⟨∩ ∩ 𝑦, 𝐵⟩))
42		eleq1 2829	. . . . . . . 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 3654	. . . . . 6 ⊢ (∩ ∩ 𝑦 ∈ V → (∃𝑥(𝑥 = ∩ ∩ 𝑦 ∧ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴)) ↔ (𝑦 = ⟨∩ ∩ 𝑦, 𝐵⟩ ∧ ∩ ∩ 𝑦 ∈ 𝐴)))
45	39, 44	bitrid 283	. . . . 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 575	. . 3 ⊢ ((𝑦 ∈ (𝐴 × {𝐵}) ∧ 𝑥 = ∩ ∩ 𝑦) ↔ ((𝑦 = ⟨∩ ∩ 𝑦, 𝐵⟩ ∧ ∩ ∩ 𝑦 ∈ 𝐴) ∧ 𝑥 = ∩ ∩ 𝑦))
48	32	adantr 480	. . . . 5 ⊢ ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) → 𝑥 = ∩ ∩ 𝑦)
49	48	pm4.71i 559	. . . 4 ⊢ ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) ↔ ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = ∩ ∩ 𝑦))
50	43	pm5.32ri 575	. . . 4 ⊢ (((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = ∩ ∩ 𝑦) ↔ ((𝑦 = ⟨∩ ∩ 𝑦, 𝐵⟩ ∧ ∩ ∩ 𝑦 ∈ 𝐴) ∧ 𝑥 = ∩ ∩ 𝑦))
51	49, 50	bitr2i 276	. . 3 ⊢ (((𝑦 = ⟨∩ ∩ 𝑦, 𝐵⟩ ∧ ∩ ∩ 𝑦 ∈ 𝐴) ∧ 𝑥 = ∩ ∩ 𝑦) ↔ (𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴))
52		ancom 460	. . 3 ⊢ ((𝑦 = ⟨𝑥, 𝐵⟩ ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = ⟨𝑥, 𝐵⟩))
53	47, 51, 52	3bitri 297	. 2 ⊢ ((𝑦 ∈ (𝐴 × {𝐵}) ∧ 𝑥 = ∩ ∩ 𝑦) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = ⟨𝑥, 𝐵⟩))
54	3, 1, 14, 16, 53	en2i 9030	1 ⊢ (𝐴 × {𝐵}) ≈ 𝐴

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  Vcvv 3480  {csn 4626  ⟨cop 4632  ∩ cint 4946   class class class wbr 5143   × cxp 5683   ≈ cen 8982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-en 8986

This theorem is referenced by:  xpsneng  9096  endisj  9098  infxpenlem  10053  hashxplem  14472  rexpen  16264  heiborlem3  37820