Theorem enex 6032

Description: The equinumerosity relationship is a set. (Contributed by SF, 23-Feb-2015.)

Assertion

Ref	Expression
enex	⊢ ≈ ∈ V

Proof of Theorem enex

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

Step	Hyp	Ref	Expression
1		df-en 6030	. . 3 ⊢ ≈ = {⟨x, y⟩ ∣ ∃f f:x–1-1-onto→y}
2		elrn2 4898	. . . . 5 ⊢ (⟨x, y⟩ ∈ ran ( Fns ⊗ ran (◡Image Swap ⊗ Fns )) ↔ ∃f⟨f, ⟨x, y⟩⟩ ∈ ( Fns ⊗ ran (◡Image Swap ⊗ Fns )))
3		df-br 4641	. . . . . . . . 9 ⊢ (f Fns x ↔ ⟨f, x⟩ ∈ Fns )
4		vex 2863	. . . . . . . . . 10 ⊢ f ∈ V
5	4	brfns 5834	. . . . . . . . 9 ⊢ (f Fns x ↔ f Fn x)
6	3, 5	bitr3i 242	. . . . . . . 8 ⊢ (⟨f, x⟩ ∈ Fns ↔ f Fn x)
7		elrn2 4898	. . . . . . . . 9 ⊢ (⟨f, y⟩ ∈ ran (◡Image Swap ⊗ Fns ) ↔ ∃g⟨g, ⟨f, y⟩⟩ ∈ (◡Image Swap ⊗ Fns ))
8		oteltxp 5783	. . . . . . . . . . . 12 ⊢ (⟨g, ⟨f, y⟩⟩ ∈ (◡Image Swap ⊗ Fns ) ↔ (⟨g, f⟩ ∈ ◡Image Swap ∧ ⟨g, y⟩ ∈ Fns ))
9		opelcnv 4894	. . . . . . . . . . . . . 14 ⊢ (⟨g, f⟩ ∈ ◡Image Swap ↔ ⟨f, g⟩ ∈ Image Swap )
10		dfcnv2 5101	. . . . . . . . . . . . . . . 16 ⊢ ◡f = ( Swap “ f)
11	10	eqeq2i 2363	. . . . . . . . . . . . . . 15 ⊢ (g = ◡f ↔ g = ( Swap “ f))
12		vex 2863	. . . . . . . . . . . . . . . 16 ⊢ g ∈ V
13	4, 12	brimage 5794	. . . . . . . . . . . . . . 15 ⊢ (fImage Swap g ↔ g = ( Swap “ f))
14		df-br 4641	. . . . . . . . . . . . . . 15 ⊢ (fImage Swap g ↔ ⟨f, g⟩ ∈ Image Swap )
15	11, 13, 14	3bitr2ri 265	. . . . . . . . . . . . . 14 ⊢ (⟨f, g⟩ ∈ Image Swap ↔ g = ◡f)
16	9, 15	bitri 240	. . . . . . . . . . . . 13 ⊢ (⟨g, f⟩ ∈ ◡Image Swap ↔ g = ◡f)
17		df-br 4641	. . . . . . . . . . . . . 14 ⊢ (g Fns y ↔ ⟨g, y⟩ ∈ Fns )
18	12	brfns 5834	. . . . . . . . . . . . . 14 ⊢ (g Fns y ↔ g Fn y)
19	17, 18	bitr3i 242	. . . . . . . . . . . . 13 ⊢ (⟨g, y⟩ ∈ Fns ↔ g Fn y)
20	16, 19	anbi12i 678	. . . . . . . . . . . 12 ⊢ ((⟨g, f⟩ ∈ ◡Image Swap ∧ ⟨g, y⟩ ∈ Fns ) ↔ (g = ◡f ∧ g Fn y))
21	8, 20	bitri 240	. . . . . . . . . . 11 ⊢ (⟨g, ⟨f, y⟩⟩ ∈ (◡Image Swap ⊗ Fns ) ↔ (g = ◡f ∧ g Fn y))
22	21	exbii 1582	. . . . . . . . . 10 ⊢ (∃g⟨g, ⟨f, y⟩⟩ ∈ (◡Image Swap ⊗ Fns ) ↔ ∃g(g = ◡f ∧ g Fn y))
23	4	cnvex 5103	. . . . . . . . . . 11 ⊢ ◡f ∈ V
24		fneq1 5174	. . . . . . . . . . 11 ⊢ (g = ◡f → (g Fn y ↔ ◡f Fn y))
25	23, 24	ceqsexv 2895	. . . . . . . . . 10 ⊢ (∃g(g = ◡f ∧ g Fn y) ↔ ◡f Fn y)
26	22, 25	bitri 240	. . . . . . . . 9 ⊢ (∃g⟨g, ⟨f, y⟩⟩ ∈ (◡Image Swap ⊗ Fns ) ↔ ◡f Fn y)
27	7, 26	bitri 240	. . . . . . . 8 ⊢ (⟨f, y⟩ ∈ ran (◡Image Swap ⊗ Fns ) ↔ ◡f Fn y)
28	6, 27	anbi12i 678	. . . . . . 7 ⊢ ((⟨f, x⟩ ∈ Fns ∧ ⟨f, y⟩ ∈ ran (◡Image Swap ⊗ Fns )) ↔ (f Fn x ∧ ◡f Fn y))
29		oteltxp 5783	. . . . . . 7 ⊢ (⟨f, ⟨x, y⟩⟩ ∈ ( Fns ⊗ ran (◡Image Swap ⊗ Fns )) ↔ (⟨f, x⟩ ∈ Fns ∧ ⟨f, y⟩ ∈ ran (◡Image Swap ⊗ Fns )))
30		dff1o4 5295	. . . . . . 7 ⊢ (f:x–1-1-onto→y ↔ (f Fn x ∧ ◡f Fn y))
31	28, 29, 30	3bitr4i 268	. . . . . 6 ⊢ (⟨f, ⟨x, y⟩⟩ ∈ ( Fns ⊗ ran (◡Image Swap ⊗ Fns )) ↔ f:x–1-1-onto→y)
32	31	exbii 1582	. . . . 5 ⊢ (∃f⟨f, ⟨x, y⟩⟩ ∈ ( Fns ⊗ ran (◡Image Swap ⊗ Fns )) ↔ ∃f f:x–1-1-onto→y)
33	2, 32	bitri 240	. . . 4 ⊢ (⟨x, y⟩ ∈ ran ( Fns ⊗ ran (◡Image Swap ⊗ Fns )) ↔ ∃f f:x–1-1-onto→y)
34	33	opabbi2i 4867	. . 3 ⊢ ran ( Fns ⊗ ran (◡Image Swap ⊗ Fns )) = {⟨x, y⟩ ∣ ∃f f:x–1-1-onto→y}
35	1, 34	eqtr4i 2376	. 2 ⊢ ≈ = ran ( Fns ⊗ ran (◡Image Swap ⊗ Fns ))
36		fnsex 5833	. . . 4 ⊢ Fns ∈ V
37		swapex 4743	. . . . . . . 8 ⊢ Swap ∈ V
38	37	imageex 5802	. . . . . . 7 ⊢ Image Swap ∈ V
39	38	cnvex 5103	. . . . . 6 ⊢ ◡Image Swap ∈ V
40	39, 36	txpex 5786	. . . . 5 ⊢ (◡Image Swap ⊗ Fns ) ∈ V
41	40	rnex 5108	. . . 4 ⊢ ran (◡Image Swap ⊗ Fns ) ∈ V
42	36, 41	txpex 5786	. . 3 ⊢ ( Fns ⊗ ran (◡Image Swap ⊗ Fns )) ∈ V
43	42	rnex 5108	. 2 ⊢ ran ( Fns ⊗ ran (◡Image Swap ⊗ Fns )) ∈ V
44	35, 43	eqeltri 2423	1 ⊢ ≈ ∈ V

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860  ⟨cop 4562  {copab 4623   class class class wbr 4640   Swap cswap 4719   “ cima 4723  ^◡ccnv 4772  ran crn 4774   Fn wfn 4777  –1-1-onto→wf1o 4781   ⊗ ctxp 5736  Imagecimage 5754   Fns cfns 5762   ≈ cen 6029

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-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-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-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-2nd 4798  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-en 6030

This theorem is referenced by:  ener  6040  ncsex  6112  ncex  6118  ovmuc  6131  mucex  6134  ovcelem1  6172  ceex  6175