Theorem fvfullfunlem3 5864

Description: Lemma for fvfullfun 5865. Part one of the full function definition agrees with the set itself over its domain. (Contributed by SF, 9-Mar-2015.)

Assertion

Ref	Expression
fvfullfunlem3	⊢ (A ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) → ((( I ∘ F) ∖ ( ∼ I ∘ F)) ‘A) = (F ‘A))

Proof of Theorem fvfullfunlem3

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

Step	Hyp	Ref	Expression
1		dffun2 5120	. . . 4 ⊢ (Fun (F ↾ dom (( I ∘ F) ∖ ( ∼ I ∘ F))) ↔ ∀x∀y∀z((x(F ↾ dom (( I ∘ F) ∖ ( ∼ I ∘ F)))y ∧ x(F ↾ dom (( I ∘ F) ∖ ( ∼ I ∘ F)))z) → y = z))
2		brres 4950	. . . . . . 7 ⊢ (x(F ↾ dom (( I ∘ F) ∖ ( ∼ I ∘ F)))y ↔ (xFy ∧ x ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F))))
3		fvfullfunlem1 5862	. . . . . . . . 9 ⊢ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) = {x ∣ ∃!z xFz}
4	3	abeq2i 2461	. . . . . . . 8 ⊢ (x ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) ↔ ∃!z xFz)
5	4	anbi2i 675	. . . . . . 7 ⊢ ((xFy ∧ x ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F))) ↔ (xFy ∧ ∃!z xFz))
6	2, 5	bitri 240	. . . . . 6 ⊢ (x(F ↾ dom (( I ∘ F) ∖ ( ∼ I ∘ F)))y ↔ (xFy ∧ ∃!z xFz))
7		brres 4950	. . . . . . 7 ⊢ (x(F ↾ dom (( I ∘ F) ∖ ( ∼ I ∘ F)))z ↔ (xFz ∧ x ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F))))
8		fvfullfunlem1 5862	. . . . . . . . 9 ⊢ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) = {x ∣ ∃!y xFy}
9	8	abeq2i 2461	. . . . . . . 8 ⊢ (x ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) ↔ ∃!y xFy)
10	9	anbi2i 675	. . . . . . 7 ⊢ ((xFz ∧ x ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F))) ↔ (xFz ∧ ∃!y xFy))
11	7, 10	bitri 240	. . . . . 6 ⊢ (x(F ↾ dom (( I ∘ F) ∖ ( ∼ I ∘ F)))z ↔ (xFz ∧ ∃!y xFy))
12		tz6.12-1 5345	. . . . . . . . 9 ⊢ ((xFy ∧ ∃!y xFy) → (F ‘x) = y)
13	12	adantrl 696	. . . . . . . 8 ⊢ ((xFy ∧ (xFz ∧ ∃!y xFy)) → (F ‘x) = y)
14		tz6.12-1 5345	. . . . . . . . 9 ⊢ ((xFz ∧ ∃!y xFy) → (F ‘x) = z)
15	14	adantl 452	. . . . . . . 8 ⊢ ((xFy ∧ (xFz ∧ ∃!y xFy)) → (F ‘x) = z)
16	13, 15	eqtr3d 2387	. . . . . . 7 ⊢ ((xFy ∧ (xFz ∧ ∃!y xFy)) → y = z)
17	16	adantlr 695	. . . . . 6 ⊢ (((xFy ∧ ∃!z xFz) ∧ (xFz ∧ ∃!y xFy)) → y = z)
18	6, 11, 17	syl2anb 465	. . . . 5 ⊢ ((x(F ↾ dom (( I ∘ F) ∖ ( ∼ I ∘ F)))y ∧ x(F ↾ dom (( I ∘ F) ∖ ( ∼ I ∘ F)))z) → y = z)
19	18	gen2 1547	. . . 4 ⊢ ∀y∀z((x(F ↾ dom (( I ∘ F) ∖ ( ∼ I ∘ F)))y ∧ x(F ↾ dom (( I ∘ F) ∖ ( ∼ I ∘ F)))z) → y = z)
20	1, 19	mpgbir 1550	. . 3 ⊢ Fun (F ↾ dom (( I ∘ F) ∖ ( ∼ I ∘ F)))
21		fvfullfunlem2 5863	. . . . 5 ⊢ (( I ∘ F) ∖ ( ∼ I ∘ F)) ⊆ F
22		ssdmrn 5100	. . . . . 6 ⊢ (( I ∘ F) ∖ ( ∼ I ∘ F)) ⊆ (dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × ran (( I ∘ F) ∖ ( ∼ I ∘ F)))
23		ssv 3292	. . . . . . 7 ⊢ ran (( I ∘ F) ∖ ( ∼ I ∘ F)) ⊆ V
24		xpss2 4858	. . . . . . 7 ⊢ (ran (( I ∘ F) ∖ ( ∼ I ∘ F)) ⊆ V → (dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × ran (( I ∘ F) ∖ ( ∼ I ∘ F))) ⊆ (dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × V))
25	23, 24	ax-mp 5	. . . . . 6 ⊢ (dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × ran (( I ∘ F) ∖ ( ∼ I ∘ F))) ⊆ (dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × V)
26	22, 25	sstri 3282	. . . . 5 ⊢ (( I ∘ F) ∖ ( ∼ I ∘ F)) ⊆ (dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × V)
27	21, 26	ssini 3479	. . . 4 ⊢ (( I ∘ F) ∖ ( ∼ I ∘ F)) ⊆ (F ∩ (dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × V))
28		df-res 4789	. . . 4 ⊢ (F ↾ dom (( I ∘ F) ∖ ( ∼ I ∘ F))) = (F ∩ (dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × V))
29	27, 28	sseqtr4i 3305	. . 3 ⊢ (( I ∘ F) ∖ ( ∼ I ∘ F)) ⊆ (F ↾ dom (( I ∘ F) ∖ ( ∼ I ∘ F)))
30		funssfv 5344	. . 3 ⊢ ((Fun (F ↾ dom (( I ∘ F) ∖ ( ∼ I ∘ F))) ∧ (( I ∘ F) ∖ ( ∼ I ∘ F)) ⊆ (F ↾ dom (( I ∘ F) ∖ ( ∼ I ∘ F))) ∧ A ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F))) → ((F ↾ dom (( I ∘ F) ∖ ( ∼ I ∘ F))) ‘A) = ((( I ∘ F) ∖ ( ∼ I ∘ F)) ‘A))
31	20, 29, 30	mp3an12 1267	. 2 ⊢ (A ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) → ((F ↾ dom (( I ∘ F) ∖ ( ∼ I ∘ F))) ‘A) = ((( I ∘ F) ∖ ( ∼ I ∘ F)) ‘A))
32		fvres 5343	. 2 ⊢ (A ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) → ((F ↾ dom (( I ∘ F) ∖ ( ∼ I ∘ F))) ‘A) = (F ‘A))
33	31, 32	eqtr3d 2387	1 ⊢ (A ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) → ((( I ∘ F) ∖ ( ∼ I ∘ F)) ‘A) = (F ‘A))

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Vcvv 2860   ∼ ccompl 3206   ∖ cdif 3207   ∩ cin 3209   ⊆ wss 3258   class class class wbr 4640   ∘ ccom 4722   I cid 4764   × cxp 4771  dom cdm 4773  ran crn 4774   ↾ cres 4775  Fun wfun 4776   ‘cfv 4782

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-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fv 4796

This theorem is referenced by:  fvfullfun  5865