Theorem enprmap 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.)

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 6078	. . . 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 6079	. . 3 ⊢ ((x ≠ y ∧ A = {x, y}) → Fun ◡(r ∈ (A ↑m B) ↦ (◡r “ {x})))
5		enprmap.1	. . . 4 ⊢ B ∈ V
6	1, 5	enprmaplem6 6082	. . 3 ⊢ ((x ≠ y ∧ A = {x, y}) → ran (r ∈ (A ↑m B) ↦ (◡r “ {x})) = ℘B)
7		dff1o2 5292	. . 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 6077	. . 3 ⊢ (r ∈ (A ↑m B) ↦ (◡r “ {x})) ∈ V
10	9	f1oen 6034	. 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 2517  Vcvv 2860  ℘cpw 3723  {csn 3738  {cpr 3739   class class class wbr 4640   “ cima 4723  ^◡ccnv 4772  ran crn 4774  Fun wfun 4776   Fn wfn 4777  –1-1-onto→wf1o 4781  (class class class)co 5526   ↦ cmpt 5652   ↑_m cmap 6000   ≈ 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-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-map 6002  df-en 6030

This theorem is referenced by:  enprmapc  6084