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 e. _V
enprmapc.2	|- B e. _V
enprmapc.3	|- C e. _V

Assertion

Ref	Expression
enprmapc	|- ((A =/= B /\ P = {A, B}) -> (P ^m C) ~~ ~PC)

Proof of Theorem enprmapc

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

Step	Hyp	Ref	Expression
1		enprmapc.1	. 2 |- A e. _V
2		neeq1 2524	. . . 4 |- (x = A -> (x =/= B <-> A =/= B))
3		preq1 3799	. . . . 5 |- (x = A -> {x, B} = {A, B})
4	3	eqeq2d 2364	. . . 4 |- (x = A -> (P = {x, B} <-> P = {A, B}))
5	2, 4	anbi12d 691	. . 3 |- (x = A -> ((x =/= B /\ P = {x, B}) <-> (A =/= B /\ P = {A, B})))
6	5	imbi1d 308	. 2 |- (x = A -> (((x =/= B /\ P = {x, B}) -> (P ^m C) ~~ ~PC) <-> ((A =/= B /\ P = {A, B}) -> (P ^m C) ~~ ~PC)))
7		enprmapc.2	. . 3 |- B e. _V
8		neeq2 2525	. . . . 5 |- (y = B -> (x =/= y <-> x =/= B))
9		preq2 3800	. . . . . 6 |- (y = B -> {x, y} = {x, B})
10	9	eqeq2d 2364	. . . . 5 |- (y = B -> (P = {x, y} <-> P = {x, B}))
11	8, 10	anbi12d 691	. . . 4 |- (y = B -> ((x =/= y /\ P = {x, y}) <-> (x =/= B /\ P = {x, B})))
12	11	imbi1d 308	. . 3 |- (y = B -> (((x =/= y /\ P = {x, y}) -> (P ^m C) ~~ ~PC) <-> ((x =/= B /\ P = {x, B}) -> (P ^m C) ~~ ~PC)))
13		enprmapc.3	. . . 4 |- C e. _V
14	13	enprmap 6082	. . 3 |- ((x =/= y /\ P = {x, y}) -> (P ^m C) ~~ ~PC)
15	7, 12, 14	vtocl 2909	. 2 |- ((x =/= B /\ P = {x, B}) -> (P ^m C) ~~ ~PC)
16	1, 6, 15	vtocl 2909	1 |- ((A =/= B /\ P = {A, B}) -> (P ^m C) ~~ ~PC)

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710   =/= wne 2516  _Vcvv 2859  ~Pcpw 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