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Theorem brtxpsd 31985
Description: Expansion of a common form used in quantifier-free definitions. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brtxpsd.1 𝐴 ∈ V
brtxpsd.2 𝐵 ∈ V
Assertion
Ref Expression
brtxpsd 𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵 ↔ ∀𝑥(𝑥𝐵𝑥𝑅𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅

Proof of Theorem brtxpsd
StepHypRef Expression
1 df-br 4652 . . 3 (𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ran ((V ⊗ E ) △ (𝑅 ⊗ V)))
2 opex 4930 . . . . 5 𝐴, 𝐵⟩ ∈ V
32elrn 5364 . . . 4 (⟨𝐴, 𝐵⟩ ∈ ran ((V ⊗ E ) △ (𝑅 ⊗ V)) ↔ ∃𝑥 𝑥((V ⊗ E ) △ (𝑅 ⊗ V))⟨𝐴, 𝐵⟩)
4 brsymdif 4709 . . . . . 6 (𝑥((V ⊗ E ) △ (𝑅 ⊗ V))⟨𝐴, 𝐵⟩ ↔ ¬ (𝑥(V ⊗ E )⟨𝐴, 𝐵⟩ ↔ 𝑥(𝑅 ⊗ V)⟨𝐴, 𝐵⟩))
5 brv 4939 . . . . . . . . 9 𝑥V𝐴
6 vex 3201 . . . . . . . . . 10 𝑥 ∈ V
7 brtxpsd.1 . . . . . . . . . 10 𝐴 ∈ V
8 brtxpsd.2 . . . . . . . . . 10 𝐵 ∈ V
96, 7, 8brtxp 31971 . . . . . . . . 9 (𝑥(V ⊗ E )⟨𝐴, 𝐵⟩ ↔ (𝑥V𝐴𝑥 E 𝐵))
105, 9mpbiran 953 . . . . . . . 8 (𝑥(V ⊗ E )⟨𝐴, 𝐵⟩ ↔ 𝑥 E 𝐵)
118epelc 5029 . . . . . . . 8 (𝑥 E 𝐵𝑥𝐵)
1210, 11bitri 264 . . . . . . 7 (𝑥(V ⊗ E )⟨𝐴, 𝐵⟩ ↔ 𝑥𝐵)
13 brv 4939 . . . . . . . 8 𝑥V𝐵
146, 7, 8brtxp 31971 . . . . . . . 8 (𝑥(𝑅 ⊗ V)⟨𝐴, 𝐵⟩ ↔ (𝑥𝑅𝐴𝑥V𝐵))
1513, 14mpbiran2 954 . . . . . . 7 (𝑥(𝑅 ⊗ V)⟨𝐴, 𝐵⟩ ↔ 𝑥𝑅𝐴)
1612, 15bibi12i 329 . . . . . 6 ((𝑥(V ⊗ E )⟨𝐴, 𝐵⟩ ↔ 𝑥(𝑅 ⊗ V)⟨𝐴, 𝐵⟩) ↔ (𝑥𝐵𝑥𝑅𝐴))
174, 16xchbinx 324 . . . . 5 (𝑥((V ⊗ E ) △ (𝑅 ⊗ V))⟨𝐴, 𝐵⟩ ↔ ¬ (𝑥𝐵𝑥𝑅𝐴))
1817exbii 1773 . . . 4 (∃𝑥 𝑥((V ⊗ E ) △ (𝑅 ⊗ V))⟨𝐴, 𝐵⟩ ↔ ∃𝑥 ¬ (𝑥𝐵𝑥𝑅𝐴))
193, 18bitri 264 . . 3 (⟨𝐴, 𝐵⟩ ∈ ran ((V ⊗ E ) △ (𝑅 ⊗ V)) ↔ ∃𝑥 ¬ (𝑥𝐵𝑥𝑅𝐴))
20 exnal 1753 . . 3 (∃𝑥 ¬ (𝑥𝐵𝑥𝑅𝐴) ↔ ¬ ∀𝑥(𝑥𝐵𝑥𝑅𝐴))
211, 19, 203bitrri 287 . 2 (¬ ∀𝑥(𝑥𝐵𝑥𝑅𝐴) ↔ 𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵)
2221con1bii 346 1 𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵 ↔ ∀𝑥(𝑥𝐵𝑥𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wal 1480  wex 1703  wcel 1989  Vcvv 3198  csymdif 3841  cop 4181   class class class wbr 4651   E cep 5026  ran crn 5113  ctxp 31921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-symdif 3842  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-eprel 5027  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fo 5892  df-fv 5894  df-1st 7165  df-2nd 7166  df-txp 31945
This theorem is referenced by:  brtxpsd2  31986
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