New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  iss GIF version

Theorem iss 5000
 Description: A subclass of the identity function is the identity function restricted to its domain. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 13-Dec-2003.) (Revised by set.mm contributors, 27-Aug-2011.)
Assertion
Ref Expression
iss (A I ↔ A = ( I dom A))

Proof of Theorem iss
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3267 . . . . . 6 (A I → (x, y Ax, y I ))
2 opeldm 4910 . . . . . . 7 (x, y Ax dom A)
32a1i 10 . . . . . 6 (A I → (x, y Ax dom A))
41, 3jcad 519 . . . . 5 (A I → (x, y A → (x, y I x dom A)))
5 df-br 4640 . . . . . . . 8 (x I yx, y I )
6 vex 2862 . . . . . . . . 9 y V
76ideq 4870 . . . . . . . 8 (x I yx = y)
85, 7bitr3i 242 . . . . . . 7 (x, y I ↔ x = y)
98anbi1i 676 . . . . . 6 ((x, y I x dom A) ↔ (x = y x dom A))
10 eldm2 4899 . . . . . . . . 9 (x dom Ayx, y A)
111, 8syl6ib 217 . . . . . . . . . . 11 (A I → (x, y Ax = y))
12 opeq2 4579 . . . . . . . . . . . . 13 (x = yx, x = x, y)
1312eleq1d 2419 . . . . . . . . . . . 12 (x = y → (x, x Ax, y A))
1413biimprd 214 . . . . . . . . . . 11 (x = y → (x, y Ax, x A))
1511, 14syli 33 . . . . . . . . . 10 (A I → (x, y Ax, x A))
1615exlimdv 1636 . . . . . . . . 9 (A I → (yx, y Ax, x A))
1710, 16syl5bi 208 . . . . . . . 8 (A I → (x dom Ax, x A))
1813biimpd 198 . . . . . . . 8 (x = y → (x, x Ax, y A))
1917, 18syl9 66 . . . . . . 7 (A I → (x = y → (x dom Ax, y A)))
2019imp3a 420 . . . . . 6 (A I → ((x = y x dom A) → x, y A))
219, 20syl5bi 208 . . . . 5 (A I → ((x, y I x dom A) → x, y A))
224, 21impbid 183 . . . 4 (A I → (x, y A ↔ (x, y I x dom A)))
23 opelres 4950 . . . 4 (x, y ( I dom A) ↔ (x, y I x dom A))
2422, 23syl6bbr 254 . . 3 (A I → (x, y Ax, y ( I dom A)))
2524eqrelrdv 4852 . 2 (A I → A = ( I dom A))
26 resss 4988 . . 3 ( I dom A) I
27 sseq1 3292 . . 3 (A = ( I dom A) → (A I ↔ ( I dom A) I ))
2826, 27mpbiri 224 . 2 (A = ( I dom A) → A I )
2925, 28impbii 180 1 (A I ↔ A = ( I dom A))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ⊆ wss 3257  ⟨cop 4561   class class class wbr 4639   I cid 4763  dom cdm 4772   ↾ cres 4774 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-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788 This theorem is referenced by:  f1ococnv2  5309
 Copyright terms: Public domain W3C validator