Theorem dfxrn2 38752

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 38749	. . 3 ⊢ Rel (𝑅 ⋉ 𝑆)
2		dfrel4v 6141	. . 3 ⊢ (Rel (𝑅 ⋉ 𝑆) ↔ (𝑅 ⋉ 𝑆) = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧})
3	1, 2	mpbi 231	. 2 ⊢ (𝑅 ⋉ 𝑆) = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧}
4		breq2 5076	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩))
5		brxrn2 38751	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
6	5	elv 3436	. . . . 5 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
7		brxrn 38750	. . . . . . . . 9 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
8	7	el3v 3439	. . . . . . . 8 ⊢ (𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
9	8	anbi2i 629	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
10		3anass 1100	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)))
11	9, 10	bitr4i 279	. . . . . 6 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
12	11	2exbii 1856	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦))
13	4	copsex2gb 5749	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩) ↔ (𝑧 ∈ (V × V) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑧))
14	6, 12, 13	3bitr2i 300	. . . 4 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 ↔ (𝑧 ∈ (V × V) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑧))
15	14	simplbi 497	. . 3 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑧 → 𝑧 ∈ (V × V))
16	4, 15	cnvoprab 8002	. 2 ⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩} = {⟨𝑢, 𝑧⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)𝑧}
17	8	oprabbii 7423	. . 3 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩} = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}
18	17	cnveqi 5816	. 2 ⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ 𝑢(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩} = ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}
19	3, 16, 18	3eqtr2i 2768	1 ⊢ (𝑅 ⋉ 𝑆) = ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}

Syntax hints:   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119  Vcvv 3431  ⟨cop 4561   class class class wbr 5072  {copab 5134   × cxp 5616  ^◡ccnv 5617  Rel wrel 5623  {coprab 7357   ⋉ cxrn 38541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fo 6491  df-fv 6493  df-oprab 7360  df-1st 7931  df-2nd 7932  df-xrn 38747

This theorem is referenced by:  dmxrn  38754