Theorem nfco 4882

Description: Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)

Hypotheses

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

Assertion

Ref	Expression
nfco	⊢ Ⅎx(A ∘ B)

Proof of Theorem nfco

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

Step	Hyp	Ref	Expression
1		df-co 4726	. 2 ⊢ (A ∘ B) = {⟨y, z⟩ ∣ ∃w(yBw ∧ wAz)}
2		nfcv 2489	. . . . . 6 ⊢ Ⅎxy
3		nfco.2	. . . . . 6 ⊢ ℲxB
4		nfcv 2489	. . . . . 6 ⊢ Ⅎxw
5	2, 3, 4	nfbr 4683	. . . . 5 ⊢ Ⅎx yBw
6		nfco.1	. . . . . 6 ⊢ ℲxA
7		nfcv 2489	. . . . . 6 ⊢ Ⅎxz
8	4, 6, 7	nfbr 4683	. . . . 5 ⊢ Ⅎx wAz
9	5, 8	nfan 1824	. . . 4 ⊢ Ⅎx(yBw ∧ wAz)
10	9	nfex 1843	. . 3 ⊢ Ⅎx∃w(yBw ∧ wAz)
11	10	nfopab 4627	. 2 ⊢ Ⅎx{⟨y, z⟩ ∣ ∃w(yBw ∧ wAz)}
12	1, 11	nfcxfr 2486	1 ⊢ Ⅎx(A ∘ B)

Syntax hints:   ∧ wa 358  ∃wex 1541  Ⅎwnfc 2476  {copab 4622   class class class wbr 4639   ∘ ccom 4721

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623  df-br 4640  df-co 4726

This theorem is referenced by:  nffun  5130