Theorem ffnfvf 5429

Description: A function maps to a class to which all values belong. This version of ffnfv 5428 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.)

Hypotheses

Ref	Expression
ffnfvf.1	|- F/_xA
ffnfvf.2	|- F/_xB
ffnfvf.3	|- F/_xF

Assertion

Ref	Expression
ffnfvf	|- (F:A-->B <-> (F Fn A /\ A.x e. A (F` x) e. B))

Proof of Theorem ffnfvf

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffnfv 5428	. 2 |- (F:A-->B <-> (F Fn A /\ A.z e. A (F` z) e. B))
2		nfcv 2490	. . . 4 |- F/_zA
3		ffnfvf.1	. . . 4 |- F/_xA
4		ffnfvf.3	. . . . . 6 |- F/_xF
5		nfcv 2490	. . . . . 6 |- F/_xz
6	4, 5	nffv 5335	. . . . 5 |- F/_x(F` z)
7		ffnfvf.2	. . . . 5 |- F/_xB
8	6, 7	nfel 2498	. . . 4 |- F/x(F` z) e. B
9		nfv 1619	. . . 4 |- F/z(F` x) e. B
10		fveq2 5329	. . . . 5 |- (z = x -> (F` z) = (F` x))
11	10	eleq1d 2419	. . . 4 |- (z = x -> ((F` z) e. B <-> (F` x) e. B))
12	2, 3, 8, 9, 11	cbvralf 2830	. . 3 |- (A.z e. A (F` z) e. B <-> A.x e. A (F` x) e. B)
13	12	anbi2i 675	. 2 |- ((F Fn A /\ A.z e. A (F` z) e. B) <-> (F Fn A /\ A.x e. A (F` x) e. B))
14	1, 13	bitri 240	1 |- (F:A-->B <-> (F Fn A /\ A.x e. A (F` x) e. B))

Syntax hints:   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710   F/_wnfc 2477  A.wral 2615   Fn wfn 4777  -->wf 4778  ` 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-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-fv 4796

This theorem is referenced by: (None)