Theorem dminxp 6172

Description: Two ways to express totality of a restricted and corestricted binary relation (intersection of a binary relation with a Cartesian product). (Contributed by NM, 17-Jan-2006.)

Assertion

Ref	Expression
dminxp	⊢ (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥𝐶𝑦)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem dminxp

Step	Hyp	Ref	Expression
1		dfdm4 5888	. . . 4 ⊢ dom (𝐶 ∩ (𝐴 × 𝐵)) = ran ◡(𝐶 ∩ (𝐴 × 𝐵))
2		cnvin 6137	. . . . . 6 ⊢ ◡(𝐶 ∩ (𝐴 × 𝐵)) = (◡𝐶 ∩ ◡(𝐴 × 𝐵))
3		cnvxp 6149	. . . . . . 7 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
4	3	ineq2i 4204	. . . . . 6 ⊢ (◡𝐶 ∩ ◡(𝐴 × 𝐵)) = (◡𝐶 ∩ (𝐵 × 𝐴))
5	2, 4	eqtri 2754	. . . . 5 ⊢ ◡(𝐶 ∩ (𝐴 × 𝐵)) = (◡𝐶 ∩ (𝐵 × 𝐴))
6	5	rneqi 5929	. . . 4 ⊢ ran ◡(𝐶 ∩ (𝐴 × 𝐵)) = ran (◡𝐶 ∩ (𝐵 × 𝐴))
7	1, 6	eqtri 2754	. . 3 ⊢ dom (𝐶 ∩ (𝐴 × 𝐵)) = ran (◡𝐶 ∩ (𝐵 × 𝐴))
8	7	eqeq1i 2731	. 2 ⊢ (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ran (◡𝐶 ∩ (𝐵 × 𝐴)) = 𝐴)
9		rninxp 6171	. 2 ⊢ (ran (◡𝐶 ∩ (𝐵 × 𝐴)) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦◡𝐶𝑥)
10		vex 3472	. . . . 5 ⊢ 𝑦 ∈ V
11		vex 3472	. . . . 5 ⊢ 𝑥 ∈ V
12	10, 11	brcnv 5875	. . . 4 ⊢ (𝑦◡𝐶𝑥 ↔ 𝑥𝐶𝑦)
13	12	rexbii 3088	. . 3 ⊢ (∃𝑦 ∈ 𝐵 𝑦◡𝐶𝑥 ↔ ∃𝑦 ∈ 𝐵 𝑥𝐶𝑦)
14	13	ralbii 3087	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦◡𝐶𝑥 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥𝐶𝑦)
15	8, 9, 14	3bitri 297	1 ⊢ (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥𝐶𝑦)

Syntax hints:   ↔ wb 205   = wceq 1533  ∀wral 3055  ∃wrex 3064   ∩ cin 3942   class class class wbr 5141   × cxp 5667  ^◡ccnv 5668  dom cdm 5669  ran crn 5670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682

This theorem is referenced by:  trust  24085  onsupmaxb  42545