Theorem nfima 4954

Description: Bound-variable hypothesis builder for image. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Hypotheses

Ref	Expression
nfima.1	⊢ ℲxA
nfima.2	⊢ ℲxB

Assertion

Ref	Expression
nfima	⊢ Ⅎx(A “ B)

Proof of Theorem nfima

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

Step	Hyp	Ref	Expression
1		df-ima 4728	. 2 ⊢ (A “ B) = {y ∣ ∃z ∈ B zAy}
2		nfima.2	. . . 4 ⊢ ℲxB
3		nfcv 2490	. . . . 5 ⊢ Ⅎxz
4		nfima.1	. . . . 5 ⊢ ℲxA
5		nfcv 2490	. . . . 5 ⊢ Ⅎxy
6	3, 4, 5	nfbr 4684	. . . 4 ⊢ Ⅎx zAy
7	2, 6	nfrex 2670	. . 3 ⊢ Ⅎx∃z ∈ B zAy
8	7	nfab 2494	. 2 ⊢ Ⅎx{y ∣ ∃z ∈ B zAy}
9	1, 8	nfcxfr 2487	1 ⊢ Ⅎx(A “ B)

Syntax hints:  {cab 2339  Ⅎwnfc 2477  ∃wrex 2616   class class class wbr 4640   “ cima 4723

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-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-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  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-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-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-br 4641  df-ima 4728

This theorem is referenced by:  nfimad  4955  csbima12g  4956  nfrn  4964