Theorem enmap2 6069

Description: Set exponentiation preserves equinumerosity in the second argument. Theorem XI.1.22 of [Rosser] p. 357. (Contributed by SF, 26-Feb-2015.)

Assertion

Ref	Expression
enmap2	⊢ (A ≈ B → (C ↑m A) ≈ (C ↑m B))

Proof of Theorem enmap2

Dummy variables a b r s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brex 4690	. 2 ⊢ (A ≈ B → (A ∈ V ∧ B ∈ V))
2		breq1 4643	. . . 4 ⊢ (a = A → (a ≈ b ↔ A ≈ b))
3		oveq2 5532	. . . . 5 ⊢ (a = A → (C ↑m a) = (C ↑m A))
4	3	breq1d 4650	. . . 4 ⊢ (a = A → ((C ↑m a) ≈ (C ↑m b) ↔ (C ↑m A) ≈ (C ↑m b)))
5	2, 4	imbi12d 311	. . 3 ⊢ (a = A → ((a ≈ b → (C ↑m a) ≈ (C ↑m b)) ↔ (A ≈ b → (C ↑m A) ≈ (C ↑m b))))
6		breq2 4644	. . . 4 ⊢ (b = B → (A ≈ b ↔ A ≈ B))
7		oveq2 5532	. . . . 5 ⊢ (b = B → (C ↑m b) = (C ↑m B))
8	7	breq2d 4652	. . . 4 ⊢ (b = B → ((C ↑m A) ≈ (C ↑m b) ↔ (C ↑m A) ≈ (C ↑m B)))
9	6, 8	imbi12d 311	. . 3 ⊢ (b = B → ((A ≈ b → (C ↑m A) ≈ (C ↑m b)) ↔ (A ≈ B → (C ↑m A) ≈ (C ↑m B))))
10		bren 6031	. . . 4 ⊢ (a ≈ b ↔ ∃r r:a–1-1-onto→b)
11		eqid 2353	. . . . . . . . . 10 ⊢ (s ∈ (C ↑m a) ↦ (s ∘ ◡r)) = (s ∈ (C ↑m a) ↦ (s ∘ ◡r))
12	11	enmap2lem4 6067	. . . . . . . . 9 ⊢ (r:a–1-1-onto→b → Fun ◡(s ∈ (C ↑m a) ↦ (s ∘ ◡r)))
13		dfrn4 4905	. . . . . . . . . 10 ⊢ ran (s ∈ (C ↑m a) ↦ (s ∘ ◡r)) = dom ◡(s ∈ (C ↑m a) ↦ (s ∘ ◡r))
14	11	enmap2lem5 6068	. . . . . . . . . 10 ⊢ (r:a–1-1-onto→b → ran (s ∈ (C ↑m a) ↦ (s ∘ ◡r)) = (C ↑m b))
15	13, 14	syl5eqr 2399	. . . . . . . . 9 ⊢ (r:a–1-1-onto→b → dom ◡(s ∈ (C ↑m a) ↦ (s ∘ ◡r)) = (C ↑m b))
16	12, 15	jca 518	. . . . . . . 8 ⊢ (r:a–1-1-onto→b → (Fun ◡(s ∈ (C ↑m a) ↦ (s ∘ ◡r)) ∧ dom ◡(s ∈ (C ↑m a) ↦ (s ∘ ◡r)) = (C ↑m b)))
17		df-fn 4791	. . . . . . . 8 ⊢ (◡(s ∈ (C ↑m a) ↦ (s ∘ ◡r)) Fn (C ↑m b) ↔ (Fun ◡(s ∈ (C ↑m a) ↦ (s ∘ ◡r)) ∧ dom ◡(s ∈ (C ↑m a) ↦ (s ∘ ◡r)) = (C ↑m b)))
18	16, 17	sylibr 203	. . . . . . 7 ⊢ (r:a–1-1-onto→b → ◡(s ∈ (C ↑m a) ↦ (s ∘ ◡r)) Fn (C ↑m b))
19	11	enmap2lem2 6065	. . . . . . . 8 ⊢ (s ∈ (C ↑m a) ↦ (s ∘ ◡r)) Fn (C ↑m a)
20		dff1o4 5295	. . . . . . . 8 ⊢ ((s ∈ (C ↑m a) ↦ (s ∘ ◡r)):(C ↑m a)–1-1-onto→(C ↑m b) ↔ ((s ∈ (C ↑m a) ↦ (s ∘ ◡r)) Fn (C ↑m a) ∧ ◡(s ∈ (C ↑m a) ↦ (s ∘ ◡r)) Fn (C ↑m b)))
21	19, 20	mpbiran 884	. . . . . . 7 ⊢ ((s ∈ (C ↑m a) ↦ (s ∘ ◡r)):(C ↑m a)–1-1-onto→(C ↑m b) ↔ ◡(s ∈ (C ↑m a) ↦ (s ∘ ◡r)) Fn (C ↑m b))
22	18, 21	sylibr 203	. . . . . 6 ⊢ (r:a–1-1-onto→b → (s ∈ (C ↑m a) ↦ (s ∘ ◡r)):(C ↑m a)–1-1-onto→(C ↑m b))
23	11	enmap2lem1 6064	. . . . . . 7 ⊢ (s ∈ (C ↑m a) ↦ (s ∘ ◡r)) ∈ V
24	23	f1oen 6034	. . . . . 6 ⊢ ((s ∈ (C ↑m a) ↦ (s ∘ ◡r)):(C ↑m a)–1-1-onto→(C ↑m b) → (C ↑m a) ≈ (C ↑m b))
25	22, 24	syl 15	. . . . 5 ⊢ (r:a–1-1-onto→b → (C ↑m a) ≈ (C ↑m b))
26	25	exlimiv 1634	. . . 4 ⊢ (∃r r:a–1-1-onto→b → (C ↑m a) ≈ (C ↑m b))
27	10, 26	sylbi 187	. . 3 ⊢ (a ≈ b → (C ↑m a) ≈ (C ↑m b))
28	5, 9, 27	vtocl2g 2919	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (A ≈ B → (C ↑m A) ≈ (C ↑m B)))
29	1, 28	mpcom 32	1 ⊢ (A ≈ B → (C ↑m A) ≈ (C ↑m B))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860   class class class wbr 4640   ∘ ccom 4722  ^◡ccnv 4772  dom cdm 4773  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 proof depends on 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 proof 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-compose 5749  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 used by:  enpw  6088  cenc  6182