Theorem dminxp 6169

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 5875	. . . 4 ⊢ dom (𝐶 ∩ (𝐴 × 𝐵)) = ran ◡(𝐶 ∩ (𝐴 × 𝐵))
2		cnvin 6133	. . . . . 6 ⊢ ◡(𝐶 ∩ (𝐴 × 𝐵)) = (◡𝐶 ∩ ◡(𝐴 × 𝐵))
3		cnvxp 6146	. . . . . . 7 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
4	3	ineq2i 4192	. . . . . 6 ⊢ (◡𝐶 ∩ ◡(𝐴 × 𝐵)) = (◡𝐶 ∩ (𝐵 × 𝐴))
5	2, 4	eqtri 2758	. . . . 5 ⊢ ◡(𝐶 ∩ (𝐴 × 𝐵)) = (◡𝐶 ∩ (𝐵 × 𝐴))
6	5	rneqi 5917	. . . 4 ⊢ ran ◡(𝐶 ∩ (𝐴 × 𝐵)) = ran (◡𝐶 ∩ (𝐵 × 𝐴))
7	1, 6	eqtri 2758	. . 3 ⊢ dom (𝐶 ∩ (𝐴 × 𝐵)) = ran (◡𝐶 ∩ (𝐵 × 𝐴))
8	7	eqeq1i 2740	. 2 ⊢ (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ran (◡𝐶 ∩ (𝐵 × 𝐴)) = 𝐴)
9		rninxp 6168	. 2 ⊢ (ran (◡𝐶 ∩ (𝐵 × 𝐴)) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦◡𝐶𝑥)
10		vex 3463	. . . . 5 ⊢ 𝑦 ∈ V
11		vex 3463	. . . . 5 ⊢ 𝑥 ∈ V
12	10, 11	brcnv 5862	. . . 4 ⊢ (𝑦◡𝐶𝑥 ↔ 𝑥𝐶𝑦)
13	12	rexbii 3083	. . 3 ⊢ (∃𝑦 ∈ 𝐵 𝑦◡𝐶𝑥 ↔ ∃𝑦 ∈ 𝐵 𝑥𝐶𝑦)
14	13	ralbii 3082	. 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 206   = wceq 1540  ∀wral 3051  ∃wrex 3060   ∩ cin 3925   class class class wbr 5119   × cxp 5652  ^◡ccnv 5653  dom cdm 5654  ran crn 5655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667

This theorem is referenced by:  trust  24168  onsupmaxb  43263