Theorem enprmap 6082

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.)

Hypothesis

Ref	Expression
enprmap.1	⊢ B ∈ V

Assertion

Ref	Expression
enprmap	⊢ ((x ≠ y ∧ A = {x, y}) → (A ↑m B) ≈ ℘B)

Proof of Theorem enprmap

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2353	. . . . 5 ⊢ (r ∈ (A ↑m B) ↦ (◡r “ {x})) = (r ∈ (A ↑m B) ↦ (◡r “ {x}))
2	1	enprmaplem2 6077	. . . 4 ⊢ (r ∈ (A ↑m B) ↦ (◡r “ {x})) Fn (A ↑m B)
3	2	a1i 10	. . 3 ⊢ ((x ≠ y ∧ A = {x, y}) → (r ∈ (A ↑m B) ↦ (◡r “ {x})) Fn (A ↑m B))
4	1	enprmaplem3 6078	. . 3 ⊢ ((x ≠ y ∧ A = {x, y}) → Fun ◡(r ∈ (A ↑m B) ↦ (◡r “ {x})))
5		enprmap.1	. . . 4 ⊢ B ∈ V
6	1, 5	enprmaplem6 6081	. . 3 ⊢ ((x ≠ y ∧ A = {x, y}) → ran (r ∈ (A ↑m B) ↦ (◡r “ {x})) = ℘B)
7		dff1o2 5291	. . 3 ⊢ ((r ∈ (A ↑m B) ↦ (◡r “ {x})):(A ↑m B)–1-1-onto→℘B ↔ ((r ∈ (A ↑m B) ↦ (◡r “ {x})) Fn (A ↑m B) ∧ Fun ◡(r ∈ (A ↑m B) ↦ (◡r “ {x})) ∧ ran (r ∈ (A ↑m B) ↦ (◡r “ {x})) = ℘B))
8	3, 4, 6, 7	syl3anbrc 1136	. 2 ⊢ ((x ≠ y ∧ A = {x, y}) → (r ∈ (A ↑m B) ↦ (◡r “ {x})):(A ↑m B)–1-1-onto→℘B)
9	1	enprmaplem1 6076	. . 3 ⊢ (r ∈ (A ↑m B) ↦ (◡r “ {x})) ∈ V
10	9	f1oen 6033	. 2 ⊢ ((r ∈ (A ↑m B) ↦ (◡r “ {x})):(A ↑m B)–1-1-onto→℘B → (A ↑m B) ≈ ℘B)
11	8, 10	syl 15	1 ⊢ ((x ≠ y ∧ A = {x, y}) → (A ↑m B) ≈ ℘B)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  Vcvv 2859  ℘cpw 3722  {csn 3737  {cpr 3738   class class class wbr 4639   “ cima 4722  ^◡ccnv 4771  ran crn 4773  Fun wfun 4775   Fn wfn 4776  –1-1-onto→wf1o 4780  (class class class)co 5525   ↦ cmpt 5651   ↑_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:  enprmapc  6083