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Theorem enmap2 6068
 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 (AB → (Cm A) ≈ (Cm B))

Proof of Theorem enmap2
Dummy variables a b r s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brex 4689 . 2 (AB → (A V B V))
2 breq1 4642 . . . 4 (a = A → (abAb))
3 oveq2 5531 . . . . 5 (a = A → (Cm a) = (Cm A))
43breq1d 4649 . . . 4 (a = A → ((Cm a) ≈ (Cm b) ↔ (Cm A) ≈ (Cm b)))
52, 4imbi12d 311 . . 3 (a = A → ((ab → (Cm a) ≈ (Cm b)) ↔ (Ab → (Cm A) ≈ (Cm b))))
6 breq2 4643 . . . 4 (b = B → (AbAB))
7 oveq2 5531 . . . . 5 (b = B → (Cm b) = (Cm B))
87breq2d 4651 . . . 4 (b = B → ((Cm A) ≈ (Cm b) ↔ (Cm A) ≈ (Cm B)))
96, 8imbi12d 311 . . 3 (b = B → ((Ab → (Cm A) ≈ (Cm b)) ↔ (AB → (Cm A) ≈ (Cm B))))
10 bren 6030 . . . 4 (abr r:a1-1-ontob)
11 eqid 2353 . . . . . . . . . 10 (s (Cm a) (s r)) = (s (Cm a) (s r))
1211enmap2lem4 6066 . . . . . . . . 9 (r:a1-1-ontob → Fun (s (Cm a) (s r)))
13 dfrn4 4904 . . . . . . . . . 10 ran (s (Cm a) (s r)) = dom (s (Cm a) (s r))
1411enmap2lem5 6067 . . . . . . . . . 10 (r:a1-1-ontob → ran (s (Cm a) (s r)) = (Cm b))
1513, 14syl5eqr 2399 . . . . . . . . 9 (r:a1-1-ontob → dom (s (Cm a) (s r)) = (Cm b))
1612, 15jca 518 . . . . . . . 8 (r:a1-1-ontob → (Fun (s (Cm a) (s r)) dom (s (Cm a) (s r)) = (Cm b)))
17 df-fn 4790 . . . . . . . 8 ((s (Cm a) (s r)) Fn (Cm b) ↔ (Fun (s (Cm a) (s r)) dom (s (Cm a) (s r)) = (Cm b)))
1816, 17sylibr 203 . . . . . . 7 (r:a1-1-ontob(s (Cm a) (s r)) Fn (Cm b))
1911enmap2lem2 6064 . . . . . . . 8 (s (Cm a) (s r)) Fn (Cm a)
20 dff1o4 5294 . . . . . . . 8 ((s (Cm a) (s r)):(Cm a)–1-1-onto→(Cm b) ↔ ((s (Cm a) (s r)) Fn (Cm a) (s (Cm a) (s r)) Fn (Cm b)))
2119, 20mpbiran 884 . . . . . . 7 ((s (Cm a) (s r)):(Cm a)–1-1-onto→(Cm b) ↔ (s (Cm a) (s r)) Fn (Cm b))
2218, 21sylibr 203 . . . . . 6 (r:a1-1-ontob → (s (Cm a) (s r)):(Cm a)–1-1-onto→(Cm b))
2311enmap2lem1 6063 . . . . . . 7 (s (Cm a) (s r)) V
2423f1oen 6033 . . . . . 6 ((s (Cm a) (s r)):(Cm a)–1-1-onto→(Cm b) → (Cm a) ≈ (Cm b))
2522, 24syl 15 . . . . 5 (r:a1-1-ontob → (Cm a) ≈ (Cm b))
2625exlimiv 1634 . . . 4 (r r:a1-1-ontob → (Cm a) ≈ (Cm b))
2710, 26sylbi 187 . . 3 (ab → (Cm a) ≈ (Cm b))
285, 9, 27vtocl2g 2918 . 2 ((A V B V) → (AB → (Cm A) ≈ (Cm B)))
291, 28mpcom 32 1 (AB → (Cm A) ≈ (Cm B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   class class class wbr 4639   ∘ ccom 4721  ◡ccnv 4771  dom cdm 4772  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-1 5  ax-2 6  ax-3 7  ax-mp 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-compose 5748  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  cenc  6181
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