Theorem dfrnf 4963

Description: Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
dfrnf.1	⊢ ℲxA
dfrnf.2	⊢ ℲyA

Assertion

Ref	Expression
dfrnf	⊢ ran A = {y ∣ ∃x xAy}

Distinct variable group:   x,y

Allowed substitution hints:   A(x,y)

Proof of Theorem dfrnf

Dummy variables w v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfrn2 4903	. 2 ⊢ ran A = {w ∣ ∃v vAw}
2		nfcv 2490	. . . . 5 ⊢ Ⅎxv
3		dfrnf.1	. . . . 5 ⊢ ℲxA
4		nfcv 2490	. . . . 5 ⊢ Ⅎxw
5	2, 3, 4	nfbr 4684	. . . 4 ⊢ Ⅎx vAw
6		nfv 1619	. . . 4 ⊢ Ⅎv xAw
7		breq1 4643	. . . 4 ⊢ (v = x → (vAw ↔ xAw))
8	5, 6, 7	cbvex 1985	. . 3 ⊢ (∃v vAw ↔ ∃x xAw)
9	8	abbii 2466	. 2 ⊢ {w ∣ ∃v vAw} = {w ∣ ∃x xAw}
10		nfcv 2490	. . . . 5 ⊢ Ⅎyx
11		dfrnf.2	. . . . 5 ⊢ ℲyA
12		nfcv 2490	. . . . 5 ⊢ Ⅎyw
13	10, 11, 12	nfbr 4684	. . . 4 ⊢ Ⅎy xAw
14	13	nfex 1843	. . 3 ⊢ Ⅎy∃x xAw
15		nfv 1619	. . 3 ⊢ Ⅎw∃x xAy
16		breq2 4644	. . . 4 ⊢ (w = y → (xAw ↔ xAy))
17	16	exbidv 1626	. . 3 ⊢ (w = y → (∃x xAw ↔ ∃x xAy))
18	14, 15, 17	cbvab 2472	. 2 ⊢ {w ∣ ∃x xAw} = {y ∣ ∃x xAy}
19	1, 9, 18	3eqtri 2377	1 ⊢ ran A = {y ∣ ∃x xAy}

Syntax hints:  ∃wex 1541   = wceq 1642  {cab 2339  Ⅎwnfc 2477   class class class wbr 4640  ran crn 4774

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-br 4641  df-ima 4728  df-rn 4787

This theorem is referenced by:  rnopab  4968