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Theorem enprmapc 6083
 Description: A mapping from a two element pair onto a set is equinumerous with the power class of the set. Theorem XI.1.28 of [Rosser] p. 360. (Contributed by SF, 3-Mar-2015.)
Hypotheses
Ref Expression
enprmapc.1 A V
enprmapc.2 B V
enprmapc.3 C V
Assertion
Ref Expression
enprmapc ((AB P = {A, B}) → (Pm C) ≈ C)

Proof of Theorem enprmapc
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 enprmapc.1 . 2 A V
2 neeq1 2524 . . . 4 (x = A → (xBAB))
3 preq1 3799 . . . . 5 (x = A → {x, B} = {A, B})
43eqeq2d 2364 . . . 4 (x = A → (P = {x, B} ↔ P = {A, B}))
52, 4anbi12d 691 . . 3 (x = A → ((xB P = {x, B}) ↔ (AB P = {A, B})))
65imbi1d 308 . 2 (x = A → (((xB P = {x, B}) → (Pm C) ≈ C) ↔ ((AB P = {A, B}) → (Pm C) ≈ C)))
7 enprmapc.2 . . 3 B V
8 neeq2 2525 . . . . 5 (y = B → (xyxB))
9 preq2 3800 . . . . . 6 (y = B → {x, y} = {x, B})
109eqeq2d 2364 . . . . 5 (y = B → (P = {x, y} ↔ P = {x, B}))
118, 10anbi12d 691 . . . 4 (y = B → ((xy P = {x, y}) ↔ (xB P = {x, B})))
1211imbi1d 308 . . 3 (y = B → (((xy P = {x, y}) → (Pm C) ≈ C) ↔ ((xB P = {x, B}) → (Pm C) ≈ C)))
13 enprmapc.3 . . . 4 C V
1413enprmap 6082 . . 3 ((xy P = {x, y}) → (Pm C) ≈ C)
157, 12, 14vtocl 2909 . 2 ((xB P = {x, B}) → (Pm C) ≈ C)
161, 6, 15vtocl 2909 1 ((AB P = {A, B}) → (Pm C) ≈ C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  Vcvv 2859  ℘cpw 3722  {cpr 3738   class class class wbr 4639  (class class class)co 5525   ↑m cmap 5999   ≈ cen 6028 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 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-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-map 6001  df-en 6029 This theorem is referenced by:  enpw  6087  ce2  6192
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