Theorem opeliunxp 4821

Description: Membership in a union of Cartesian products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.)

Assertion

Ref	Expression
opeliunxp	⊢ (⟨x, C⟩ ∈ ∪x ∈ A ({x} × B) ↔ (x ∈ A ∧ C ∈ B))

Proof of Theorem opeliunxp

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

Step	Hyp	Ref	Expression
1		elex 2868	. . 3 ⊢ (⟨x, C⟩ ∈ ∪x ∈ A ({x} × B) → ⟨x, C⟩ ∈ V)
2		opexb 4604	. . . 4 ⊢ (⟨x, C⟩ ∈ V ↔ (x ∈ V ∧ C ∈ V))
3	2	simprbi 450	. . 3 ⊢ (⟨x, C⟩ ∈ V → C ∈ V)
4	1, 3	syl 15	. 2 ⊢ (⟨x, C⟩ ∈ ∪x ∈ A ({x} × B) → C ∈ V)
5		elex 2868	. . 3 ⊢ (C ∈ B → C ∈ V)
6	5	adantl 452	. 2 ⊢ ((x ∈ A ∧ C ∈ B) → C ∈ V)
7		vex 2863	. . . . 5 ⊢ x ∈ V
8		opexg 4588	. . . . 5 ⊢ ((x ∈ V ∧ C ∈ V) → ⟨x, C⟩ ∈ V)
9	7, 8	mpan 651	. . . 4 ⊢ (C ∈ V → ⟨x, C⟩ ∈ V)
10		df-rex 2621	. . . . . . 7 ⊢ (∃x ∈ A y ∈ ({x} × B) ↔ ∃x(x ∈ A ∧ y ∈ ({x} × B)))
11		nfv 1619	. . . . . . . 8 ⊢ Ⅎz(x ∈ A ∧ y ∈ ({x} × B))
12		nfs1v 2106	. . . . . . . . 9 ⊢ Ⅎx[z / x]x ∈ A
13		nfcv 2490	. . . . . . . . . . 11 ⊢ Ⅎx{z}
14		nfcsb1v 3169	. . . . . . . . . . 11 ⊢ Ⅎx[z / x]B
15	13, 14	nfxp 4811	. . . . . . . . . 10 ⊢ Ⅎx({z} × [z / x]B)
16	15	nfcri 2484	. . . . . . . . 9 ⊢ Ⅎx y ∈ ({z} × [z / x]B)
17	12, 16	nfan 1824	. . . . . . . 8 ⊢ Ⅎx([z / x]x ∈ A ∧ y ∈ ({z} × [z / x]B))
18		sbequ12 1919	. . . . . . . . 9 ⊢ (x = z → (x ∈ A ↔ [z / x]x ∈ A))
19		sneq 3745	. . . . . . . . . . 11 ⊢ (x = z → {x} = {z})
20		csbeq1a 3145	. . . . . . . . . . 11 ⊢ (x = z → B = [z / x]B)
21	19, 20	xpeq12d 4810	. . . . . . . . . 10 ⊢ (x = z → ({x} × B) = ({z} × [z / x]B))
22	21	eleq2d 2420	. . . . . . . . 9 ⊢ (x = z → (y ∈ ({x} × B) ↔ y ∈ ({z} × [z / x]B)))
23	18, 22	anbi12d 691	. . . . . . . 8 ⊢ (x = z → ((x ∈ A ∧ y ∈ ({x} × B)) ↔ ([z / x]x ∈ A ∧ y ∈ ({z} × [z / x]B))))
24	11, 17, 23	cbvex 1985	. . . . . . 7 ⊢ (∃x(x ∈ A ∧ y ∈ ({x} × B)) ↔ ∃z([z / x]x ∈ A ∧ y ∈ ({z} × [z / x]B)))
25	10, 24	bitri 240	. . . . . 6 ⊢ (∃x ∈ A y ∈ ({x} × B) ↔ ∃z([z / x]x ∈ A ∧ y ∈ ({z} × [z / x]B)))
26		eleq1 2413	. . . . . . . 8 ⊢ (y = ⟨x, C⟩ → (y ∈ ({z} × [z / x]B) ↔ ⟨x, C⟩ ∈ ({z} × [z / x]B)))
27	26	anbi2d 684	. . . . . . 7 ⊢ (y = ⟨x, C⟩ → (([z / x]x ∈ A ∧ y ∈ ({z} × [z / x]B)) ↔ ([z / x]x ∈ A ∧ ⟨x, C⟩ ∈ ({z} × [z / x]B))))
28	27	exbidv 1626	. . . . . 6 ⊢ (y = ⟨x, C⟩ → (∃z([z / x]x ∈ A ∧ y ∈ ({z} × [z / x]B)) ↔ ∃z([z / x]x ∈ A ∧ ⟨x, C⟩ ∈ ({z} × [z / x]B))))
29	25, 28	syl5bb 248	. . . . 5 ⊢ (y = ⟨x, C⟩ → (∃x ∈ A y ∈ ({x} × B) ↔ ∃z([z / x]x ∈ A ∧ ⟨x, C⟩ ∈ ({z} × [z / x]B))))
30		df-iun 3972	. . . . 5 ⊢ ∪x ∈ A ({x} × B) = {y ∣ ∃x ∈ A y ∈ ({x} × B)}
31	29, 30	elab2g 2988	. . . 4 ⊢ (⟨x, C⟩ ∈ V → (⟨x, C⟩ ∈ ∪x ∈ A ({x} × B) ↔ ∃z([z / x]x ∈ A ∧ ⟨x, C⟩ ∈ ({z} × [z / x]B))))
32	9, 31	syl 15	. . 3 ⊢ (C ∈ V → (⟨x, C⟩ ∈ ∪x ∈ A ({x} × B) ↔ ∃z([z / x]x ∈ A ∧ ⟨x, C⟩ ∈ ({z} × [z / x]B))))
33		opelxp 4812	. . . . . . 7 ⊢ (⟨x, C⟩ ∈ ({z} × [z / x]B) ↔ (x ∈ {z} ∧ C ∈ [z / x]B))
34	33	anbi2i 675	. . . . . 6 ⊢ (([z / x]x ∈ A ∧ ⟨x, C⟩ ∈ ({z} × [z / x]B)) ↔ ([z / x]x ∈ A ∧ (x ∈ {z} ∧ C ∈ [z / x]B)))
35		an12 772	. . . . . 6 ⊢ (([z / x]x ∈ A ∧ (x ∈ {z} ∧ C ∈ [z / x]B)) ↔ (x ∈ {z} ∧ ([z / x]x ∈ A ∧ C ∈ [z / x]B)))
36		elsn 3749	. . . . . . . 8 ⊢ (x ∈ {z} ↔ x = z)
37		equcom 1680	. . . . . . . 8 ⊢ (x = z ↔ z = x)
38	36, 37	bitri 240	. . . . . . 7 ⊢ (x ∈ {z} ↔ z = x)
39	38	anbi1i 676	. . . . . 6 ⊢ ((x ∈ {z} ∧ ([z / x]x ∈ A ∧ C ∈ [z / x]B)) ↔ (z = x ∧ ([z / x]x ∈ A ∧ C ∈ [z / x]B)))
40	34, 35, 39	3bitri 262	. . . . 5 ⊢ (([z / x]x ∈ A ∧ ⟨x, C⟩ ∈ ({z} × [z / x]B)) ↔ (z = x ∧ ([z / x]x ∈ A ∧ C ∈ [z / x]B)))
41	40	exbii 1582	. . . 4 ⊢ (∃z([z / x]x ∈ A ∧ ⟨x, C⟩ ∈ ({z} × [z / x]B)) ↔ ∃z(z = x ∧ ([z / x]x ∈ A ∧ C ∈ [z / x]B)))
42		sbequ12r 1920	. . . . . 6 ⊢ (z = x → ([z / x]x ∈ A ↔ x ∈ A))
43	20	equcoms 1681	. . . . . . . 8 ⊢ (z = x → B = [z / x]B)
44	43	eqcomd 2358	. . . . . . 7 ⊢ (z = x → [z / x]B = B)
45	44	eleq2d 2420	. . . . . 6 ⊢ (z = x → (C ∈ [z / x]B ↔ C ∈ B))
46	42, 45	anbi12d 691	. . . . 5 ⊢ (z = x → (([z / x]x ∈ A ∧ C ∈ [z / x]B) ↔ (x ∈ A ∧ C ∈ B)))
47	7, 46	ceqsexv 2895	. . . 4 ⊢ (∃z(z = x ∧ ([z / x]x ∈ A ∧ C ∈ [z / x]B)) ↔ (x ∈ A ∧ C ∈ B))
48	41, 47	bitri 240	. . 3 ⊢ (∃z([z / x]x ∈ A ∧ ⟨x, C⟩ ∈ ({z} × [z / x]B)) ↔ (x ∈ A ∧ C ∈ B))
49	32, 48	syl6bb 252	. 2 ⊢ (C ∈ V → (⟨x, C⟩ ∈ ∪x ∈ A ({x} × B) ↔ (x ∈ A ∧ C ∈ B)))
50	4, 6, 49	pm5.21nii 342	1 ⊢ (⟨x, C⟩ ∈ ∪x ∈ A ({x} × B) ↔ (x ∈ A ∧ C ∈ B))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642  [wsb 1648   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860  [csb 3137  {csn 3738  ∪ciun 3970  ⟨cop 4562   × cxp 4771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-xp 4785

This theorem is referenced by:  eliunxp  4822  opeliunxp2  4823