Theorem brtxp 36265

Description: Characterize a ternary relation over a tail Cartesian product. Together with txpss3v 36263, this completely defines membership in a tail cross. (Contributed by Scott Fenton, 31-Mar-2012.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)

Hypotheses

Ref	Expression
brtxp.1	⊢ 𝑋 ∈ V
brtxp.2	⊢ 𝑌 ∈ V
brtxp.3	⊢ 𝑍 ∈ V

Assertion

Ref	Expression
brtxp	⊢ (𝑋(𝐴 ⊗ 𝐵)⟨𝑌, 𝑍⟩ ↔ (𝑋𝐴𝑌 ∧ 𝑋𝐵𝑍))

Proof of Theorem brtxp

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

Step	Hyp	Ref	Expression
1		df-txp 36239	. . 3 ⊢ (𝐴 ⊗ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
2	1	breqi 5116	. 2 ⊢ (𝑋(𝐴 ⊗ 𝐵)⟨𝑌, 𝑍⟩ ↔ 𝑋((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))⟨𝑌, 𝑍⟩)
3		brin 5164	. 2 ⊢ (𝑋((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))⟨𝑌, 𝑍⟩ ↔ (𝑋(◡(1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ∧ 𝑋(◡(2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩))
4		brtxp.1	. . . . 5 ⊢ 𝑋 ∈ V
5		opex 5443	. . . . 5 ⊢ ⟨𝑌, 𝑍⟩ ∈ V
6	4, 5	brco 5854	. . . 4 ⊢ (𝑋(◡(1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ↔ ∃𝑦(𝑋𝐴𝑦 ∧ 𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩))
7		vex 3467	. . . . . . . 8 ⊢ 𝑦 ∈ V
8	7, 5	brcnv 5866	. . . . . . 7 ⊢ (𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ ⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦)
9		brtxp.2	. . . . . . . . 9 ⊢ 𝑌 ∈ V
10		brtxp.3	. . . . . . . . 9 ⊢ 𝑍 ∈ V
11	9, 10	opelvv 5699	. . . . . . . 8 ⊢ ⟨𝑌, 𝑍⟩ ∈ (V × V)
12	7	brresi 5985	. . . . . . . 8 ⊢ (⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦 ↔ (⟨𝑌, 𝑍⟩ ∈ (V × V) ∧ ⟨𝑌, 𝑍⟩1st 𝑦))
13	11, 12	mpbiran 721	. . . . . . 7 ⊢ (⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦 ↔ ⟨𝑌, 𝑍⟩1st 𝑦)
14	9, 10	br1steq 36158	. . . . . . 7 ⊢ (⟨𝑌, 𝑍⟩1st 𝑦 ↔ 𝑦 = 𝑌)
15	8, 13, 14	3bitri 300	. . . . . 6 ⊢ (𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ 𝑦 = 𝑌)
16	15	anbi1ci 637	. . . . 5 ⊢ ((𝑋𝐴𝑦 ∧ 𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑦 = 𝑌 ∧ 𝑋𝐴𝑦))
17	16	exbii 1875	. . . 4 ⊢ (∃𝑦(𝑋𝐴𝑦 ∧ 𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ ∃𝑦(𝑦 = 𝑌 ∧ 𝑋𝐴𝑦))
18		breq2 5114	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋𝐴𝑦 ↔ 𝑋𝐴𝑌))
19	9, 18	ceqsexv 3511	. . . 4 ⊢ (∃𝑦(𝑦 = 𝑌 ∧ 𝑋𝐴𝑦) ↔ 𝑋𝐴𝑌)
20	6, 17, 19	3bitri 300	. . 3 ⊢ (𝑋(◡(1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ↔ 𝑋𝐴𝑌)
21	4, 5	brco 5854	. . . 4 ⊢ (𝑋(◡(2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩ ↔ ∃𝑧(𝑋𝐵𝑧 ∧ 𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩))
22		vex 3467	. . . . . . . 8 ⊢ 𝑧 ∈ V
23	22, 5	brcnv 5866	. . . . . . 7 ⊢ (𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ ⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧)
24	22	brresi 5985	. . . . . . . 8 ⊢ (⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧 ↔ (⟨𝑌, 𝑍⟩ ∈ (V × V) ∧ ⟨𝑌, 𝑍⟩2nd 𝑧))
25	11, 24	mpbiran 721	. . . . . . 7 ⊢ (⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧 ↔ ⟨𝑌, 𝑍⟩2nd 𝑧)
26	9, 10	br2ndeq 36159	. . . . . . 7 ⊢ (⟨𝑌, 𝑍⟩2nd 𝑧 ↔ 𝑧 = 𝑍)
27	23, 25, 26	3bitri 300	. . . . . 6 ⊢ (𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ 𝑧 = 𝑍)
28	27	anbi1ci 637	. . . . 5 ⊢ ((𝑋𝐵𝑧 ∧ 𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑧 = 𝑍 ∧ 𝑋𝐵𝑧))
29	28	exbii 1875	. . . 4 ⊢ (∃𝑧(𝑋𝐵𝑧 ∧ 𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ ∃𝑧(𝑧 = 𝑍 ∧ 𝑋𝐵𝑧))
30		breq2 5114	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑋𝐵𝑧 ↔ 𝑋𝐵𝑍))
31	10, 30	ceqsexv 3511	. . . 4 ⊢ (∃𝑧(𝑧 = 𝑍 ∧ 𝑋𝐵𝑧) ↔ 𝑋𝐵𝑍)
32	21, 29, 31	3bitri 300	. . 3 ⊢ (𝑋(◡(2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩ ↔ 𝑋𝐵𝑍)
33	20, 32	anbi12i 639	. 2 ⊢ ((𝑋(◡(1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ∧ 𝑋(◡(2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩) ↔ (𝑋𝐴𝑌 ∧ 𝑋𝐵𝑍))
34	2, 3, 33	3bitri 300	1 ⊢ (𝑋(𝐴 ⊗ 𝐵)⟨𝑌, 𝑍⟩ ↔ (𝑋𝐴𝑌 ∧ 𝑋𝐵𝑍))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  Vcvv 3463   ∩ cin 3912  ⟨cop 4597   class class class wbr 5110   × cxp 5657  ^◡ccnv 5658   ↾ cres 5661   ∘ ccom 5663  1^st c1st 7980  2^nd c2nd 7981   ⊗ ctxp 36215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fo 6539  df-fv 6541  df-1st 7982  df-2nd 7983  df-txp 36239

This theorem is referenced by:  brtxp2  36266  pprodss4v  36269  brpprod  36270  brsset  36274  brtxpsd  36279  elfuns  36300