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Theorem enex 6031
 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.
StepHypRef Expression
1 df-en 6029 . . 3 ≈ = {x, y f f:x1-1-ontoy}
2 elrn2 4897 . . . . 5 (x, y ran ( Fns ⊗ ran (Image Swap Fns )) ↔ ff, x, y ( Fns ⊗ ran (Image Swap Fns )))
3 df-br 4640 . . . . . . . . 9 (f Fns xf, x Fns )
4 vex 2862 . . . . . . . . . 10 f V
54brfns 5833 . . . . . . . . 9 (f Fns xf Fn x)
63, 5bitr3i 242 . . . . . . . 8 (f, x Fnsf Fn x)
7 elrn2 4897 . . . . . . . . 9 (f, y ran (Image Swap Fns ) ↔ gg, f, y (Image Swap Fns ))
8 oteltxp 5782 . . . . . . . . . . . 12 (g, f, y (Image Swap Fns ) ↔ (g, f Image Swap g, y Fns ))
9 opelcnv 4893 . . . . . . . . . . . . . 14 (g, f Image Swap f, g Image Swap )
10 dfcnv2 5100 . . . . . . . . . . . . . . . 16 f = ( Swap f)
1110eqeq2i 2363 . . . . . . . . . . . . . . 15 (g = fg = ( Swap f))
12 vex 2862 . . . . . . . . . . . . . . . 16 g V
134, 12brimage 5793 . . . . . . . . . . . . . . 15 (fImage Swap gg = ( Swap f))
14 df-br 4640 . . . . . . . . . . . . . . 15 (fImage Swap gf, g Image Swap )
1511, 13, 143bitr2ri 265 . . . . . . . . . . . . . 14 (f, g Image Swap g = f)
169, 15bitri 240 . . . . . . . . . . . . 13 (g, f Image Swap g = f)
17 df-br 4640 . . . . . . . . . . . . . 14 (g Fns yg, y Fns )
1812brfns 5833 . . . . . . . . . . . . . 14 (g Fns yg Fn y)
1917, 18bitr3i 242 . . . . . . . . . . . . 13 (g, y Fnsg Fn y)
2016, 19anbi12i 678 . . . . . . . . . . . 12 ((g, f Image Swap g, y Fns ) ↔ (g = f g Fn y))
218, 20bitri 240 . . . . . . . . . . 11 (g, f, y (Image Swap Fns ) ↔ (g = f g Fn y))
2221exbii 1582 . . . . . . . . . 10 (gg, f, y (Image Swap Fns ) ↔ g(g = f g Fn y))
234cnvex 5102 . . . . . . . . . . 11 f V
24 fneq1 5173 . . . . . . . . . . 11 (g = f → (g Fn yf Fn y))
2523, 24ceqsexv 2894 . . . . . . . . . 10 (g(g = f g Fn y) ↔ f Fn y)
2622, 25bitri 240 . . . . . . . . 9 (gg, f, y (Image Swap Fns ) ↔ f Fn y)
277, 26bitri 240 . . . . . . . 8 (f, y ran (Image Swap Fns ) ↔ f Fn y)
286, 27anbi12i 678 . . . . . . 7 ((f, x Fns f, y ran (Image Swap Fns )) ↔ (f Fn x f Fn y))
29 oteltxp 5782 . . . . . . 7 (f, x, y ( Fns ⊗ ran (Image Swap Fns )) ↔ (f, x Fns f, y ran (Image Swap Fns )))
30 dff1o4 5294 . . . . . . 7 (f:x1-1-ontoy ↔ (f Fn x f Fn y))
3128, 29, 303bitr4i 268 . . . . . 6 (f, x, y ( Fns ⊗ ran (Image Swap Fns )) ↔ f:x1-1-ontoy)
3231exbii 1582 . . . . 5 (ff, x, y ( Fns ⊗ ran (Image Swap Fns )) ↔ f f:x1-1-ontoy)
332, 32bitri 240 . . . 4 (x, y ran ( Fns ⊗ ran (Image Swap Fns )) ↔ f f:x1-1-ontoy)
3433opabbi2i 4866 . . 3 ran ( Fns ⊗ ran (Image Swap Fns )) = {x, y f f:x1-1-ontoy}
351, 34eqtr4i 2376 . 2 ≈ = ran ( Fns ⊗ ran (Image Swap Fns ))
36 fnsex 5832 . . . 4 Fns V
37 swapex 4742 . . . . . . . 8 Swap V
3837imageex 5801 . . . . . . 7 Image Swap V
3938cnvex 5102 . . . . . 6 Image Swap V
4039, 36txpex 5785 . . . . 5 (Image Swap Fns ) V
4140rnex 5107 . . . 4 ran (Image Swap Fns ) V
4236, 41txpex 5785 . . 3 ( Fns ⊗ ran (Image Swap Fns )) V
4342rnex 5107 . 2 ran ( Fns ⊗ ran (Image Swap Fns )) V
4435, 43eqeltri 2423 1 V
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ⟨cop 4561  {copab 4622   class class class wbr 4639   Swap cswap 4718   “ cima 4722  ◡ccnv 4771  ran crn 4773   Fn wfn 4776  –1-1-onto→wf1o 4780   ⊗ ctxp 5735  Imagecimage 5753   Fns cfns 5761   ≈ cen 6028 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-en 6029 This theorem is referenced by:  ener  6039  ncsex  6111  ncex  6117  ovmuc  6130  mucex  6133  ovcelem1  6171  ceex  6174
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