Theorem fvunsn 5445

Description: Remove an ordered pair not participating in a function value. (Contributed by set.mm contributors, 1-Oct-2013.) (Revised by Mario Carneiro, 28-May-2014.)

Assertion

Ref	Expression
fvunsn	|- (B =/= D -> ((A u. {<.B, C>.})` D) = (A` D))

Proof of Theorem fvunsn

Step	Hyp	Ref	Expression
1		resundir 4983	. . . 4 |- ((A u. {<.B, C>.}) |` {D}) = ((A |` {D}) u. ({<.B, C>.} |` {D}))
2		elsni 3758	. . . . . . . 8 |- (B e. {D} -> B = D)
3	2	necon3ai 2557	. . . . . . 7 |- (B =/= D -> -. B e. {D})
4		ressnop0 5437	. . . . . . 7 |- (-. B e. {D} -> ({<.B, C>.} |` {D}) = (/))
5	3, 4	syl 15	. . . . . 6 |- (B =/= D -> ({<.B, C>.} |` {D}) = (/))
6	5	uneq2d 3419	. . . . 5 |- (B =/= D -> ((A |` {D}) u. ({<.B, C>.} |` {D})) = ((A |` {D}) u. (/)))
7		un0 3576	. . . . 5 |- ((A |` {D}) u. (/)) = (A |` {D})
8	6, 7	syl6eq 2401	. . . 4 |- (B =/= D -> ((A |` {D}) u. ({<.B, C>.} |` {D})) = (A |` {D}))
9	1, 8	syl5eq 2397	. . 3 |- (B =/= D -> ((A u. {<.B, C>.}) |` {D}) = (A |` {D}))
10	9	fveq1d 5331	. 2 |- (B =/= D -> (((A u. {<.B, C>.}) |` {D})` D) = ((A |` {D})` D))
11		snidg 3759	. . . 4 |- (D e. _V -> D e. {D})
12		fvres 5343	. . . 4 |- (D e. {D} -> (((A u. {<.B, C>.}) |` {D})` D) = ((A u. {<.B, C>.})` D))
13	11, 12	syl 15	. . 3 |- (D e. _V -> (((A u. {<.B, C>.}) |` {D})` D) = ((A u. {<.B, C>.})` D))
14		fvprc 5326	. . . 4 |- (-. D e. _V -> (((A u. {<.B, C>.}) |` {D})` D) = (/))
15		fvprc 5326	. . . 4 |- (-. D e. _V -> ((A u. {<.B, C>.})` D) = (/))
16	14, 15	eqtr4d 2388	. . 3 |- (-. D e. _V -> (((A u. {<.B, C>.}) |` {D})` D) = ((A u. {<.B, C>.})` D))
17	13, 16	pm2.61i 156	. 2 |- (((A u. {<.B, C>.}) |` {D})` D) = ((A u. {<.B, C>.})` D)
18		fvres 5343	. . . 4 |- (D e. {D} -> ((A |` {D})` D) = (A` D))
19	11, 18	syl 15	. . 3 |- (D e. _V -> ((A |` {D})` D) = (A` D))
20		fvprc 5326	. . . 4 |- (-. D e. _V -> ((A |` {D})` D) = (/))
21		fvprc 5326	. . . 4 |- (-. D e. _V -> (A` D) = (/))
22	20, 21	eqtr4d 2388	. . 3 |- (-. D e. _V -> ((A |` {D})` D) = (A` D))
23	19, 22	pm2.61i 156	. 2 |- ((A |` {D})` D) = (A` D)
24	10, 17, 23	3eqtr3g 2408	1 |- (B =/= D -> ((A u. {<.B, C>.})` D) = (A` D))

Syntax hints:  -. wn 3   -> wi 4   = wceq 1642   e. wcel 1710   =/= wne 2517  _Vcvv 2860   u. cun 3208  (/)c0 3551  {csn 3738  <.cop 4562   |` cres 4775  ` 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-xp 4785  df-res 4789  df-fv 4796

This theorem is referenced by:  fvpr1  5450