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Theorem pw1fnf1o 5855
 Description: Pw1Fn is a one-to-one function with domain 1c and range ℘1c. (Contributed by SF, 26-Feb-2015.)
Assertion
Ref Expression
pw1fnf1o Pw1Fn :1c1-1-onto1c

Proof of Theorem pw1fnf1o
Dummy variables a b x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnpw1fn 5853 . 2 Pw1Fn Fn 1c
2 df-pw1fn 5766 . . . 4 Pw1Fn = (x 1c 1x)
32rnmpt 5686 . . 3 ran Pw1Fn = {y x 1c y = 1x}
4 vex 2862 . . . . . . 7 y V
54sspw1 4335 . . . . . 6 (y 1V ↔ z(z V y = 1z))
6 df1c2 4168 . . . . . . 7 1c = 1V
76sseq2i 3296 . . . . . 6 (y 1cy 1V)
8 ssv 3291 . . . . . . . 8 z V
98biantrur 492 . . . . . . 7 (y = 1z ↔ (z V y = 1z))
109exbii 1582 . . . . . 6 (z y = 1zz(z V y = 1z))
115, 7, 103bitr4i 268 . . . . 5 (y 1cz y = 1z)
124elpw 3728 . . . . 5 (y 1cy 1c)
13 df-rex 2620 . . . . . 6 (x 1c y = 1xx(x 1c y = 1x))
14 el1c 4139 . . . . . . . . 9 (x 1cz x = {z})
1514anbi1i 676 . . . . . . . 8 ((x 1c y = 1x) ↔ (z x = {z} y = 1x))
16 19.41v 1901 . . . . . . . 8 (z(x = {z} y = 1x) ↔ (z x = {z} y = 1x))
1715, 16bitr4i 243 . . . . . . 7 ((x 1c y = 1x) ↔ z(x = {z} y = 1x))
1817exbii 1582 . . . . . 6 (x(x 1c y = 1x) ↔ xz(x = {z} y = 1x))
19 excom 1741 . . . . . . 7 (xz(x = {z} y = 1x) ↔ zx(x = {z} y = 1x))
20 snex 4111 . . . . . . . . 9 {z} V
21 unieq 3900 . . . . . . . . . . . 12 (x = {z} → x = {z})
22 vex 2862 . . . . . . . . . . . . 13 z V
2322unisn 3907 . . . . . . . . . . . 12 {z} = z
2421, 23syl6eq 2401 . . . . . . . . . . 11 (x = {z} → x = z)
25 pw1eq 4143 . . . . . . . . . . 11 (x = z1x = 1z)
2624, 25syl 15 . . . . . . . . . 10 (x = {z} → 1x = 1z)
2726eqeq2d 2364 . . . . . . . . 9 (x = {z} → (y = 1xy = 1z))
2820, 27ceqsexv 2894 . . . . . . . 8 (x(x = {z} y = 1x) ↔ y = 1z)
2928exbii 1582 . . . . . . 7 (zx(x = {z} y = 1x) ↔ z y = 1z)
3019, 29bitri 240 . . . . . 6 (xz(x = {z} y = 1x) ↔ z y = 1z)
3113, 18, 303bitri 262 . . . . 5 (x 1c y = 1xz y = 1z)
3211, 12, 313bitr4i 268 . . . 4 (y 1cx 1c y = 1x)
3332abbi2i 2464 . . 3 1c = {y x 1c y = 1x}
343, 33eqtr4i 2376 . 2 ran Pw1Fn = 1c
35 el1c 4139 . . . . . 6 (x 1ca x = {a})
36 el1c 4139 . . . . . 6 (y 1cb y = {b})
3735, 36anbi12i 678 . . . . 5 ((x 1c y 1c) ↔ (a x = {a} b y = {b}))
38 eeanv 1913 . . . . 5 (ab(x = {a} y = {b}) ↔ (a x = {a} b y = {b}))
3937, 38bitr4i 243 . . . 4 ((x 1c y 1c) ↔ ab(x = {a} y = {b}))
40 pw111 4170 . . . . . . . 8 (1a = 1ba = b)
4140biimpi 186 . . . . . . 7 (1a = 1ba = b)
4241a1i 10 . . . . . 6 ((x = {a} y = {b}) → (1a = 1ba = b))
43 fveq2 5328 . . . . . . . 8 (x = {a} → ( Pw1Fnx) = ( Pw1Fn ‘{a}))
44 vex 2862 . . . . . . . . 9 a V
4544pw1fnval 5851 . . . . . . . 8 ( Pw1Fn ‘{a}) = 1a
4643, 45syl6eq 2401 . . . . . . 7 (x = {a} → ( Pw1Fnx) = 1a)
47 fveq2 5328 . . . . . . . 8 (y = {b} → ( Pw1Fny) = ( Pw1Fn ‘{b}))
48 vex 2862 . . . . . . . . 9 b V
4948pw1fnval 5851 . . . . . . . 8 ( Pw1Fn ‘{b}) = 1b
5047, 49syl6eq 2401 . . . . . . 7 (y = {b} → ( Pw1Fny) = 1b)
5146, 50eqeqan12d 2368 . . . . . 6 ((x = {a} y = {b}) → (( Pw1Fnx) = ( Pw1Fny) ↔ 1a = 1b))
52 eqeq12 2365 . . . . . . 7 ((x = {a} y = {b}) → (x = y ↔ {a} = {b}))
5344sneqb 3876 . . . . . . 7 ({a} = {b} ↔ a = b)
5452, 53syl6bb 252 . . . . . 6 ((x = {a} y = {b}) → (x = ya = b))
5542, 51, 543imtr4d 259 . . . . 5 ((x = {a} y = {b}) → (( Pw1Fnx) = ( Pw1Fny) → x = y))
5655exlimivv 1635 . . . 4 (ab(x = {a} y = {b}) → (( Pw1Fnx) = ( Pw1Fny) → x = y))
5739, 56sylbi 187 . . 3 ((x 1c y 1c) → (( Pw1Fnx) = ( Pw1Fny) → x = y))
5857rgen2a 2680 . 2 x 1c y 1c (( Pw1Fnx) = ( Pw1Fny) → x = y)
59 dff1o6 5475 . 2 ( Pw1Fn :1c1-1-onto1c ↔ ( Pw1Fn Fn 1c ran Pw1Fn = 1c x 1c y 1c (( Pw1Fnx) = ( Pw1Fny) → x = y)))
601, 34, 58, 59mpbir3an 1134 1 Pw1Fn :1c1-1-onto1c
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  ∃wrex 2615  Vcvv 2859   ⊆ wss 3257  ℘cpw 3722  {csn 3737  ∪cuni 3891  1cc1c 4134  ℘1cpw1 4135  ran crn 4773   Fn wfn 4776  –1-1-onto→wf1o 4780   ‘cfv 4781   Pw1Fn cpw1fn 5765 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-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-mpt 5652  df-pw1fn 5766 This theorem is referenced by:  enpw1pw  6075
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