Theorem dfxrn2 35788

Description: Alternate definition of the range Cartesian product. (Contributed by Peter Mazsa, 20-Feb-2022.)

Assertion

Ref	Expression
dfxrn2	⊢ (𝑅 ⋉ 𝑆) = ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}

Distinct variable groups:   𝑢,𝑅,𝑥,𝑦   𝑢,𝑆,𝑥,𝑦

Proof of Theorem dfxrn2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrnrel 35785	. . 3 ⊢ Rel (𝑅 ⋉ 𝑆)
2		dfrel4v 6014	. . 3 ⊢ (Rel (𝑅 ⋉ 𝑆) ↔ (𝑅 ⋉ 𝑆) = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧})
3	1, 2	mpbi 233	. 2 ⊢ (𝑅 ⋉ 𝑆) = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧}
4		breq2 5034	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩))
5		brxrn2 35787	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
6	5	elv 3446	. . . . 5 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
7		brxrn 35786	. . . . . . . . 9 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
8	7	el3v 35651	. . . . . . . 8 ⊢ (𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
9	8	anbi2i 625	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
10		3anass 1092	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
11	9, 10	bitr4i 281	. . . . . 6 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
12	11	2exbii 1850	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
13	4	copsex2gb 5643	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 ∈ (V × V) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑧))
14	6, 12, 13	3bitr2i 302	. . . 4 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ (𝑧 ∈ (V × V) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑧))
15	14	simplbi 501	. . 3 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 → 𝑧 ∈ (V × V))
16	4, 15	cnvoprab 7740	. 2 ⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩} = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧}
17	8	oprabbii 7200	. . 3 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩} = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}
18	17	cnveqi 5709	. 2 ⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩} = ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}
19	3, 16, 18	3eqtr2i 2827	1 ⊢ (𝑅 ⋉ 𝑆) = ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}

Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3441  ⟨cop 4531   class class class wbr 5030  {copab 5092   × cxp 5517  ^◡ccnv 5518  Rel wrel 5524  {coprab 7136   ⋉ cxrn 35612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fo 6330  df-fv 6332  df-oprab 7139  df-1st 7671  df-2nd 7672  df-xrn 35783

This theorem is referenced by: (None)