Theorem fnfullfunlem1 5857

Description: Lemma for fnfullfun 5859. Binary relationship over part one of the full function definition. (Contributed by SF, 9-Mar-2015.)

Assertion

Ref	Expression
fnfullfunlem1	⊢ (A(( I ∘ F) ∖ ( ∼ I ∘ F))B ↔ (AFB ∧ ∀x(AFx → x = B)))

Distinct variable groups:   x,A   x,B   x,F

Proof of Theorem fnfullfunlem1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brex 4690	. . 3 ⊢ (A(( I ∘ F) ∖ ( ∼ I ∘ F))B → (A ∈ V ∧ B ∈ V))
2	1	simprd 449	. 2 ⊢ (A(( I ∘ F) ∖ ( ∼ I ∘ F))B → B ∈ V)
3		brex 4690	. . . 4 ⊢ (AFB → (A ∈ V ∧ B ∈ V))
4	3	simprd 449	. . 3 ⊢ (AFB → B ∈ V)
5	4	adantr 451	. 2 ⊢ ((AFB ∧ ∀x(AFx → x = B)) → B ∈ V)
6		breq2 4644	. . . 4 ⊢ (y = B → (A(( I ∘ F) ∖ ( ∼ I ∘ F))y ↔ A(( I ∘ F) ∖ ( ∼ I ∘ F))B))
7		breq2 4644	. . . . 5 ⊢ (y = B → (AFy ↔ AFB))
8		eqeq2 2362	. . . . . . 7 ⊢ (y = B → (x = y ↔ x = B))
9	8	imbi2d 307	. . . . . 6 ⊢ (y = B → ((AFx → x = y) ↔ (AFx → x = B)))
10	9	albidv 1625	. . . . 5 ⊢ (y = B → (∀x(AFx → x = y) ↔ ∀x(AFx → x = B)))
11	7, 10	anbi12d 691	. . . 4 ⊢ (y = B → ((AFy ∧ ∀x(AFx → x = y)) ↔ (AFB ∧ ∀x(AFx → x = B))))
12	6, 11	bibi12d 312	. . 3 ⊢ (y = B → ((A(( I ∘ F) ∖ ( ∼ I ∘ F))y ↔ (AFy ∧ ∀x(AFx → x = y))) ↔ (A(( I ∘ F) ∖ ( ∼ I ∘ F))B ↔ (AFB ∧ ∀x(AFx → x = B)))))
13		brdif 4695	. . . 4 ⊢ (A(( I ∘ F) ∖ ( ∼ I ∘ F))y ↔ (A( I ∘ F)y ∧ ¬ A( ∼ I ∘ F)y))
14		coi2 5096	. . . . . 6 ⊢ ( I ∘ F) = F
15	14	breqi 4646	. . . . 5 ⊢ (A( I ∘ F)y ↔ AFy)
16		brco 4884	. . . . . . 7 ⊢ (A( ∼ I ∘ F)y ↔ ∃x(AFx ∧ x ∼ I y))
17		df-br 4641	. . . . . . . . . 10 ⊢ (x ∼ I y ↔ ⟨x, y⟩ ∈ ∼ I )
18		vex 2863	. . . . . . . . . . . . 13 ⊢ x ∈ V
19		vex 2863	. . . . . . . . . . . . 13 ⊢ y ∈ V
20	18, 19	opex 4589	. . . . . . . . . . . 12 ⊢ ⟨x, y⟩ ∈ V
21	20	elcompl 3226	. . . . . . . . . . 11 ⊢ (⟨x, y⟩ ∈ ∼ I ↔ ¬ ⟨x, y⟩ ∈ I )
22		df-br 4641	. . . . . . . . . . . 12 ⊢ (x I y ↔ ⟨x, y⟩ ∈ I )
23	19	ideq 4871	. . . . . . . . . . . 12 ⊢ (x I y ↔ x = y)
24	22, 23	bitr3i 242	. . . . . . . . . . 11 ⊢ (⟨x, y⟩ ∈ I ↔ x = y)
25	21, 24	xchbinx 301	. . . . . . . . . 10 ⊢ (⟨x, y⟩ ∈ ∼ I ↔ ¬ x = y)
26	17, 25	bitri 240	. . . . . . . . 9 ⊢ (x ∼ I y ↔ ¬ x = y)
27	26	anbi2i 675	. . . . . . . 8 ⊢ ((AFx ∧ x ∼ I y) ↔ (AFx ∧ ¬ x = y))
28	27	exbii 1582	. . . . . . 7 ⊢ (∃x(AFx ∧ x ∼ I y) ↔ ∃x(AFx ∧ ¬ x = y))
29		exanali 1585	. . . . . . 7 ⊢ (∃x(AFx ∧ ¬ x = y) ↔ ¬ ∀x(AFx → x = y))
30	16, 28, 29	3bitrri 263	. . . . . 6 ⊢ (¬ ∀x(AFx → x = y) ↔ A( ∼ I ∘ F)y)
31	30	con1bii 321	. . . . 5 ⊢ (¬ A( ∼ I ∘ F)y ↔ ∀x(AFx → x = y))
32	15, 31	anbi12i 678	. . . 4 ⊢ ((A( I ∘ F)y ∧ ¬ A( ∼ I ∘ F)y) ↔ (AFy ∧ ∀x(AFx → x = y)))
33	13, 32	bitri 240	. . 3 ⊢ (A(( I ∘ F) ∖ ( ∼ I ∘ F))y ↔ (AFy ∧ ∀x(AFx → x = y)))
34	12, 33	vtoclg 2915	. 2 ⊢ (B ∈ V → (A(( I ∘ F) ∖ ( ∼ I ∘ F))B ↔ (AFB ∧ ∀x(AFx → x = B))))
35	2, 5, 34	pm5.21nii 342	1 ⊢ (A(( I ∘ F) ∖ ( ∼ I ∘ F))B ↔ (AFB ∧ ∀x(AFx → x = B)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∼ ccompl 3206   ∖ cdif 3207  ⟨cop 4562   class class class wbr 4640   ∘ ccom 4722   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-co 4727  df-id 4768  df-cnv 4786

This theorem is referenced by:  fnfullfunlem2  5858  fvfullfunlem1  5862  fvfullfunlem2  5863