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Theorem enpw1pw 6075
 Description: Unit power class and power class commute within equivalence. Theorem XI.1.35 of [Rosser] p. 368. (Contributed by SF, 26-Feb-2015.)
Hypothesis
Ref Expression
enpw1pw.1 A V
Assertion
Ref Expression
enpw1pw 1A1A

Proof of Theorem enpw1pw
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pw1fnf1o 5855 . . . . 5 Pw1Fn :1c1-1-onto1c
2 f1of1 5286 . . . . 5 ( Pw1Fn :1c1-1-onto1cPw1Fn :1c1-11c)
31, 2ax-mp 5 . . . 4 Pw1Fn :1c1-11c
4 pw1ss1c 4158 . . . 4 1A 1c
5 f1ores 5300 . . . 4 (( Pw1Fn :1c1-11c 1A 1c) → ( Pw1Fn 1A):1A1-1-onto→( Pw1Fn1A))
63, 4, 5mp2an 653 . . 3 ( Pw1Fn 1A):1A1-1-onto→( Pw1Fn1A)
7 df-ima 4727 . . . . 5 ( Pw1Fn1A) = {x y 1 Ay Pw1Fn x}
8 vex 2862 . . . . . . . . 9 x V
98elpw 3728 . . . . . . . 8 (x 1Ax 1A)
108sspw1 4335 . . . . . . . 8 (x 1Az(z A x = 1z))
11 df-rex 2620 . . . . . . . . 9 (z Ax = 1zz(z A x = 1z))
12 df-pw 3724 . . . . . . . . . . . 12 A = {z z A}
1312abeq2i 2460 . . . . . . . . . . 11 (z Az A)
1413anbi1i 676 . . . . . . . . . 10 ((z A x = 1z) ↔ (z A x = 1z))
1514exbii 1582 . . . . . . . . 9 (z(z A x = 1z) ↔ z(z A x = 1z))
1611, 15bitr2i 241 . . . . . . . 8 (z(z A x = 1z) ↔ z Ax = 1z)
179, 10, 163bitri 262 . . . . . . 7 (x 1Az Ax = 1z)
18 df-rex 2620 . . . . . . . 8 (y 1 Ay Pw1Fn xy(y 1A y Pw1Fn x))
19 elpw1 4144 . . . . . . . . . . 11 (y 1Az Ay = {z})
2019anbi1i 676 . . . . . . . . . 10 ((y 1A y Pw1Fn x) ↔ (z Ay = {z} y Pw1Fn x))
21 r19.41v 2764 . . . . . . . . . 10 (z A(y = {z} y Pw1Fn x) ↔ (z Ay = {z} y Pw1Fn x))
2220, 21bitr4i 243 . . . . . . . . 9 ((y 1A y Pw1Fn x) ↔ z A(y = {z} y Pw1Fn x))
2322exbii 1582 . . . . . . . 8 (y(y 1A y Pw1Fn x) ↔ yz A(y = {z} y Pw1Fn x))
24 rexcom4 2878 . . . . . . . . 9 (z Ay(y = {z} y Pw1Fn x) ↔ yz A(y = {z} y Pw1Fn x))
25 snex 4111 . . . . . . . . . . . 12 {z} V
26 breq1 4642 . . . . . . . . . . . 12 (y = {z} → (y Pw1Fn x ↔ {z} Pw1Fn x))
2725, 26ceqsexv 2894 . . . . . . . . . . 11 (y(y = {z} y Pw1Fn x) ↔ {z} Pw1Fn x)
28 vex 2862 . . . . . . . . . . . 12 z V
2928brpw1fn 5854 . . . . . . . . . . 11 ({z} Pw1Fn xx = 1z)
3027, 29bitri 240 . . . . . . . . . 10 (y(y = {z} y Pw1Fn x) ↔ x = 1z)
3130rexbii 2639 . . . . . . . . 9 (z Ay(y = {z} y Pw1Fn x) ↔ z Ax = 1z)
3224, 31bitr3i 242 . . . . . . . 8 (yz A(y = {z} y Pw1Fn x) ↔ z Ax = 1z)
3318, 23, 323bitri 262 . . . . . . 7 (y 1 Ay Pw1Fn xz Ax = 1z)
3417, 33bitr4i 243 . . . . . 6 (x 1Ay 1 Ay Pw1Fn x)
3534abbi2i 2464 . . . . 5 1A = {x y 1 Ay Pw1Fn x}
367, 35eqtr4i 2376 . . . 4 ( Pw1Fn1A) = 1A
37 f1oeq3 5283 . . . 4 (( Pw1Fn1A) = 1A → (( Pw1Fn 1A):1A1-1-onto→( Pw1Fn1A) ↔ ( Pw1Fn 1A):1A1-1-onto1A))
3836, 37ax-mp 5 . . 3 (( Pw1Fn 1A):1A1-1-onto→( Pw1Fn1A) ↔ ( Pw1Fn 1A):1A1-1-onto1A)
396, 38mpbi 199 . 2 ( Pw1Fn 1A):1A1-1-onto1A
40 pw1fnex 5852 . . . 4 Pw1Fn V
41 enpw1pw.1 . . . . . 6 A V
4241pwex 4329 . . . . 5 A V
4342pw1ex 4303 . . . 4 1A V
4440, 43resex 5117 . . 3 ( Pw1Fn 1A) V
4544f1oen 6033 . 2 (( Pw1Fn 1A):1A1-1-onto1A1A1A)
4639, 45ax-mp 5 1 1A1A
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  Vcvv 2859   ⊆ wss 3257  ℘cpw 3722  {csn 3737  1cc1c 4134  ℘1cpw1 4135   class class class wbr 4639   “ cima 4722   ↾ cres 4774  –1-1→wf1 4778  –1-1-onto→wf1o 4780   Pw1Fn cpw1fn 5765   ≈ 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-mpt 5652  df-txp 5736  df-ins2 5750  df-ins3 5752  df-pw1fn 5766  df-en 6029 This theorem is referenced by:  ncpwpw1  6153
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