Theorem brtxp 35868

Description: Characterize a ternary relation over a tail Cartesian product. Together with txpss3v 35866, 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 35842	. . 3 ⊢ (𝐴 ⊗ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
2	1	breqi 5113	. 2 ⊢ (𝑋(𝐴 ⊗ 𝐵)⟨𝑌, 𝑍⟩ ↔ 𝑋((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))⟨𝑌, 𝑍⟩)
3		brin 5159	. 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 5424	. . . . 5 ⊢ ⟨𝑌, 𝑍⟩ ∈ V
6	4, 5	brco 5834	. . . 4 ⊢ (𝑋(◡(1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ↔ ∃𝑦(𝑋𝐴𝑦 ∧ 𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩))
7		vex 3451	. . . . . . . 8 ⊢ 𝑦 ∈ V
8	7, 5	brcnv 5846	. . . . . . 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 5678	. . . . . . . 8 ⊢ ⟨𝑌, 𝑍⟩ ∈ (V × V)
12	7	brresi 5959	. . . . . . . 8 ⊢ (⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦 ↔ (⟨𝑌, 𝑍⟩ ∈ (V × V) ∧ ⟨𝑌, 𝑍⟩1st 𝑦))
13	11, 12	mpbiran 709	. . . . . . 7 ⊢ (⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦 ↔ ⟨𝑌, 𝑍⟩1st 𝑦)
14	9, 10	br1steq 35758	. . . . . . 7 ⊢ (⟨𝑌, 𝑍⟩1st 𝑦 ↔ 𝑦 = 𝑌)
15	8, 13, 14	3bitri 297	. . . . . 6 ⊢ (𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ 𝑦 = 𝑌)
16	15	anbi1ci 626	. . . . 5 ⊢ ((𝑋𝐴𝑦 ∧ 𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑦 = 𝑌 ∧ 𝑋𝐴𝑦))
17	16	exbii 1848	. . . 4 ⊢ (∃𝑦(𝑋𝐴𝑦 ∧ 𝑦◡(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ ∃𝑦(𝑦 = 𝑌 ∧ 𝑋𝐴𝑦))
18		breq2 5111	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋𝐴𝑦 ↔ 𝑋𝐴𝑌))
19	9, 18	ceqsexv 3498	. . . 4 ⊢ (∃𝑦(𝑦 = 𝑌 ∧ 𝑋𝐴𝑦) ↔ 𝑋𝐴𝑌)
20	6, 17, 19	3bitri 297	. . 3 ⊢ (𝑋(◡(1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ↔ 𝑋𝐴𝑌)
21	4, 5	brco 5834	. . . 4 ⊢ (𝑋(◡(2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩ ↔ ∃𝑧(𝑋𝐵𝑧 ∧ 𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩))
22		vex 3451	. . . . . . . 8 ⊢ 𝑧 ∈ V
23	22, 5	brcnv 5846	. . . . . . 7 ⊢ (𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ ⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧)
24	22	brresi 5959	. . . . . . . 8 ⊢ (⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧 ↔ (⟨𝑌, 𝑍⟩ ∈ (V × V) ∧ ⟨𝑌, 𝑍⟩2nd 𝑧))
25	11, 24	mpbiran 709	. . . . . . 7 ⊢ (⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧 ↔ ⟨𝑌, 𝑍⟩2nd 𝑧)
26	9, 10	br2ndeq 35759	. . . . . . 7 ⊢ (⟨𝑌, 𝑍⟩2nd 𝑧 ↔ 𝑧 = 𝑍)
27	23, 25, 26	3bitri 297	. . . . . 6 ⊢ (𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ 𝑧 = 𝑍)
28	27	anbi1ci 626	. . . . 5 ⊢ ((𝑋𝐵𝑧 ∧ 𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑧 = 𝑍 ∧ 𝑋𝐵𝑧))
29	28	exbii 1848	. . . 4 ⊢ (∃𝑧(𝑋𝐵𝑧 ∧ 𝑧◡(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ ∃𝑧(𝑧 = 𝑍 ∧ 𝑋𝐵𝑧))
30		breq2 5111	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑋𝐵𝑧 ↔ 𝑋𝐵𝑍))
31	10, 30	ceqsexv 3498	. . . 4 ⊢ (∃𝑧(𝑧 = 𝑍 ∧ 𝑋𝐵𝑧) ↔ 𝑋𝐵𝑍)
32	21, 29, 31	3bitri 297	. . 3 ⊢ (𝑋(◡(2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩ ↔ 𝑋𝐵𝑍)
33	20, 32	anbi12i 628	. 2 ⊢ ((𝑋(◡(1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ∧ 𝑋(◡(2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩) ↔ (𝑋𝐴𝑌 ∧ 𝑋𝐵𝑍))
34	2, 3, 33	3bitri 297	1 ⊢ (𝑋(𝐴 ⊗ 𝐵)⟨𝑌, 𝑍⟩ ↔ (𝑋𝐴𝑌 ∧ 𝑋𝐵𝑍))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3447   ∩ cin 3913  ⟨cop 4595   class class class wbr 5107   × cxp 5636  ^◡ccnv 5637   ↾ cres 5640   ∘ ccom 5642  1^st c1st 7966  2^nd c2nd 7967   ⊗ ctxp 35818

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

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-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fo 6517  df-fv 6519  df-1st 7968  df-2nd 7969  df-txp 35842

This theorem is referenced by:  brtxp2  35869  pprodss4v  35872  brpprod  35873  brsset  35877  brtxpsd  35882  elfuns  35903