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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
Assertion
Ref Expression
enpw1pw 1 1

Proof of Theorem enpw1pw
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pw1fnf1o 5855 . . . . 5 Pw1Fn 1c1c
2 f1of1 5286 . . . . 5 Pw1Fn 1c1c Pw1Fn 1c1c
31, 2ax-mp 5 . . . 4 Pw1Fn 1c1c
4 pw1ss1c 4158 . . . 4 1 1c
5 f1ores 5300 . . . 4 Pw1Fn 1c1c 1 1c Pw1Fn 1 1 Pw1Fn 1
63, 4, 5mp2an 653 . . 3 Pw1Fn 1 1 Pw1Fn 1
7 df-ima 4727 . . . . 5 Pw1Fn 1 1 Pw1Fn
8 vex 2862 . . . . . . . . 9
98elpw 3728 . . . . . . . 8 1 1
108sspw1 4335 . . . . . . . 8 1 1
11 df-rex 2620 . . . . . . . . 9 1 1
12 df-pw 3724 . . . . . . . . . . . 12
1312abeq2i 2460 . . . . . . . . . . 11
1413anbi1i 676 . . . . . . . . . 10 1 1
1514exbii 1582 . . . . . . . . 9 1 1
1611, 15bitr2i 241 . . . . . . . 8 1 1
179, 10, 163bitri 262 . . . . . . 7 1 1
18 df-rex 2620 . . . . . . . 8 1 Pw1Fn 1 Pw1Fn
19 elpw1 4144 . . . . . . . . . . 11 1
2019anbi1i 676 . . . . . . . . . 10 1 Pw1Fn Pw1Fn
21 r19.41v 2764 . . . . . . . . . 10 Pw1Fn Pw1Fn
2220, 21bitr4i 243 . . . . . . . . 9 1 Pw1Fn Pw1Fn
2322exbii 1582 . . . . . . . 8 1 Pw1Fn Pw1Fn
24 rexcom4 2878 . . . . . . . . 9 Pw1Fn Pw1Fn
25 snex 4111 . . . . . . . . . . . 12
26 breq1 4642 . . . . . . . . . . . 12 Pw1Fn Pw1Fn
2725, 26ceqsexv 2894 . . . . . . . . . . 11 Pw1Fn Pw1Fn
28 vex 2862 . . . . . . . . . . . 12
2928brpw1fn 5854 . . . . . . . . . . 11 Pw1Fn 1
3027, 29bitri 240 . . . . . . . . . 10 Pw1Fn 1
3130rexbii 2639 . . . . . . . . 9 Pw1Fn 1
3224, 31bitr3i 242 . . . . . . . 8 Pw1Fn 1
3318, 23, 323bitri 262 . . . . . . 7 1 Pw1Fn 1
3417, 33bitr4i 243 . . . . . 6 1 1 Pw1Fn
3534abbi2i 2464 . . . . 5 1 1 Pw1Fn
367, 35eqtr4i 2376 . . . 4 Pw1Fn 1 1
37 f1oeq3 5283 . . . 4 Pw1Fn 1 1 Pw1Fn 1 1 Pw1Fn 1 Pw1Fn 1 1 1
3836, 37ax-mp 5 . . 3 Pw1Fn 1 1 Pw1Fn 1 Pw1Fn 1 1 1
396, 38mpbi 199 . 2 Pw1Fn 1 1 1
40 pw1fnex 5852 . . . 4 Pw1Fn
41 enpw1pw.1 . . . . . 6
4241pwex 4329 . . . . 5
4342pw1ex 4303 . . . 4 1
4440, 43resex 5117 . . 3 Pw1Fn 1
4544f1oen 6033 . 2 Pw1Fn 1 1 1 1 1
4639, 45ax-mp 5 1 1 1
 Colors of variables: wff setvar class Syntax hints:   wb 176   wa 358  wex 1541   wceq 1642   wcel 1710  cab 2339  wrex 2615  cvv 2859   wss 3257  cpw 3722  csn 3737  1cc1c 4134  1 cpw1 4135   class class class wbr 4639  cima 4722   cres 4774  wf1 4778  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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