Theorem dfdm4 4691

Description: Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)

Assertion

Ref	Expression
dfdm4	⊢ dom 𝐴 = ran ◡𝐴

Proof of Theorem dfdm4

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

Step	Hyp	Ref	Expression
1		vex 2660	. . . . 5 ⊢ 𝑦 ∈ V
2		vex 2660	. . . . 5 ⊢ 𝑥 ∈ V
3	1, 2	brcnv 4682	. . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦)
4	3	exbii 1567	. . 3 ⊢ (∃𝑦 𝑦◡𝐴𝑥 ↔ ∃𝑦 𝑥𝐴𝑦)
5	4	abbii 2230	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦◡𝐴𝑥} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
6		dfrn2 4687	. 2 ⊢ ran ◡𝐴 = {𝑥 ∣ ∃𝑦 𝑦◡𝐴𝑥}
7		df-dm 4509	. 2 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
8	5, 6, 7	3eqtr4ri 2146	1 ⊢ dom 𝐴 = ran ◡𝐴

Syntax hints:   = wceq 1314  ∃wex 1451  {cab 2101   class class class wbr 3895  ^◡ccnv 4498  dom cdm 4499  ran crn 4500

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-cnv 4507  df-dm 4509  df-rn 4510

This theorem is referenced by:  dmcnvcnv  4723  rncnvcnv  4724  rncoeq  4770  cnvimass  4860  cnvimarndm  4861  dminxp  4941  cnvsn0  4965  rnsnopg  4975  dmmpt  4992  dmco  5005  cores2  5009  cnvssrndm  5018  cocnvres  5021  unidmrn  5029  dfdm2  5031  cnvexg  5034  funimacnv  5157  foimacnv  5341  funcocnv2  5348  fimacnv  5503  f1opw2  5930  fopwdom  6683  sbthlemi4  6800  exmidfodomrlemim  7005