Theorem brtxp2 36073

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 36070	. . . . . . 7 ⊢ (𝑅 ⊗ 𝑆) ⊆ (V × (V × V))
2	1	brel 5689	. . . . . 6 ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ (V × V)))
3	2	simprd 495	. . . . 5 ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 → 𝐵 ∈ (V × V))
4		elvv 5699	. . . . 5 ⊢ (𝐵 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
5	3, 4	sylib 218	. . . 4 ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 → ∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
6	5	pm4.71ri 560	. . 3 ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 ↔ (∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)𝐵))
7		19.41vv 1951	. . 3 ⊢ (∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)𝐵) ↔ (∃𝑥∃𝑦 𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)𝐵))
8	6, 7	bitr4i 278	. 2 ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)𝐵))
9		breq2 5102	. . . 4 ⊢ (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐴(𝑅 ⊗ 𝑆)𝐵 ↔ 𝐴(𝑅 ⊗ 𝑆)⟨𝑥, 𝑦⟩))
10	9	pm5.32i 574	. . 3 ⊢ ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)𝐵) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)⟨𝑥, 𝑦⟩))
11	10	2exbii 1850	. 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 3444	. . . . . 6 ⊢ 𝑥 ∈ V
14		vex 3444	. . . . . 6 ⊢ 𝑦 ∈ V
15	12, 13, 14	brtxp 36072	. . . . 5 ⊢ (𝐴(𝑅 ⊗ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))
16	15	anbi2i 623	. . . 4 ⊢ ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)⟨𝑥, 𝑦⟩) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
17		3anass 1094	. . . 4 ⊢ ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
18	16, 17	bitr4i 278	. . 3 ⊢ ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)⟨𝑥, 𝑦⟩) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))
19	18	2exbii 1850	. 2 ⊢ (∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅 ⊗ 𝑆)⟨𝑥, 𝑦⟩) ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))
20	8, 11, 19	3bitri 297	1 ⊢ (𝐴(𝑅 ⊗ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  Vcvv 3440  ⟨cop 4586   class class class wbr 5098   × cxp 5622   ⊗ ctxp 36022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-1st 7933  df-2nd 7934  df-txp 36046

This theorem is referenced by:  lemsuccf  36133  brrestrict  36143