Theorem imai 5011

Description: Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by set.mm contributors, 30-Apr-1998.)

Assertion

Ref	Expression
imai	⊢ ( I “ A) = A

Proof of Theorem imai

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfima4 4953	. 2 ⊢ ( I “ A) = {y ∣ ∃x(x ∈ A ∧ ⟨x, y⟩ ∈ I )}
2		df-br 4641	. . . . . . . 8 ⊢ (x I y ↔ ⟨x, y⟩ ∈ I )
3		vex 2863	. . . . . . . . 9 ⊢ y ∈ V
4	3	ideq 4871	. . . . . . . 8 ⊢ (x I y ↔ x = y)
5	2, 4	bitr3i 242	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ I ↔ x = y)
6	5	anbi2i 675	. . . . . 6 ⊢ ((x ∈ A ∧ ⟨x, y⟩ ∈ I ) ↔ (x ∈ A ∧ x = y))
7		ancom 437	. . . . . 6 ⊢ ((x ∈ A ∧ x = y) ↔ (x = y ∧ x ∈ A))
8	6, 7	bitri 240	. . . . 5 ⊢ ((x ∈ A ∧ ⟨x, y⟩ ∈ I ) ↔ (x = y ∧ x ∈ A))
9	8	exbii 1582	. . . 4 ⊢ (∃x(x ∈ A ∧ ⟨x, y⟩ ∈ I ) ↔ ∃x(x = y ∧ x ∈ A))
10		eleq1 2413	. . . . 5 ⊢ (x = y → (x ∈ A ↔ y ∈ A))
11	3, 10	ceqsexv 2895	. . . 4 ⊢ (∃x(x = y ∧ x ∈ A) ↔ y ∈ A)
12	9, 11	bitri 240	. . 3 ⊢ (∃x(x ∈ A ∧ ⟨x, y⟩ ∈ I ) ↔ y ∈ A)
13	12	abbii 2466	. 2 ⊢ {y ∣ ∃x(x ∈ A ∧ ⟨x, y⟩ ∈ I )} = {y ∣ y ∈ A}
14		abid2 2471	. 2 ⊢ {y ∣ y ∈ A} = A
15	1, 13, 14	3eqtri 2377	1 ⊢ ( I “ A) = A

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ⟨cop 4562   class class class wbr 4640   “ cima 4723   I cid 4764

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-ima 4728  df-id 4768

This theorem is referenced by:  rnresi  5012  cnvresid  5167  ecidsn  5974