Theorem brtxp2 36269

Description: The binary relation over a tail cross when the second argument is not an ordered pair. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 3-May-2015.)

Hypothesis

Ref	Expression
brtxp2.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
brtxp2	⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦

Proof of Theorem brtxp2

Step	Hyp	Ref	Expression
1		txpss3v 36266	. . . . . . 7 ⊢ (𝑅 ⊗ 𝑆) ⊆ (V × (V × V))
2	1	brel 5727	. . . . . 6 ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ (V × V)))
3	2	simprd 500	. . . . 5 ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 → 𝐵 ∈ (V × V))
4		elvv 5737	. . . . 5 ⊢ (𝐵 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
5	3, 4	sylib 221	. . . 4 ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 → ∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
6	5	pm4.71ri 569	. . 3 ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 ↔ (∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)𝐵))
7		19.41vv 1977	. . 3 ⊢ (∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)𝐵) ↔ (∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)𝐵))
8	6, 7	bitr4i 281	. 2 ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)𝐵))
9		breq2 5117	. . . 4 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐴(𝑅 ⊗ 𝑆)𝐵 ↔ 𝐴(𝑅 ⊗ 𝑆)⟨𝑥, 𝑦⟩))
10	9	pm5.32i 584	. . 3 ⊢ ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)𝐵) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)⟨𝑥, 𝑦⟩))
11	10	2exbii 1876	. 2 ⊢ (∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)𝐵) ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)⟨𝑥, 𝑦⟩))
12		brtxp2.1	. . . . . 6 ⊢ 𝐴 ∈ V
13		vex 3467	. . . . . 6 ⊢ 𝑥 ∈ V
14		vex 3467	. . . . . 6 ⊢ 𝑦 ∈ V
15	12, 13, 14	brtxp 36268	. . . . 5 ⊢ (𝐴(𝑅 ⊗ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))
16	15	anbi2i 634	. . . 4 ⊢ ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)⟨𝑥, 𝑦⟩) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
17		3anass 1109	. . . 4 ⊢ ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
18	16, 17	bitr4i 281	. . 3 ⊢ ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)⟨𝑥, 𝑦⟩) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))
19	18	2exbii 1876	. 2 ⊢ (∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)⟨𝑥, 𝑦⟩) ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))
20	8, 11, 19	3bitri 300	1 ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567  ∃wex 1806   ∈ wcel 2149  Vcvv 3463  ⟨cop 4600   class class class wbr 5113   × cxp 5660   ⊗ ctxp 36218

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 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

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 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-1st 7985  df-2nd 7986  df-txp 36242

This theorem is referenced by:  lemsuccf  36329  brrestrict  36339