ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mpoexg GIF version

Theorem mpoexg 6075
Description: Existence of an operation class abstraction (special case). (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 1-Sep-2015.)
Hypothesis
Ref Expression
mpoexg.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
mpoexg ((𝐴𝑅𝐵𝑆) → 𝐹 ∈ V)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵,𝑥
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem mpoexg
StepHypRef Expression
1 elex 2669 . . 3 (𝐵𝑆𝐵 ∈ V)
2 elex 2669 . . . 4 (𝐵 ∈ V → 𝐵 ∈ V)
32ralrimivw 2481 . . 3 (𝐵 ∈ V → ∀𝑥𝐴 𝐵 ∈ V)
41, 3syl 14 . 2 (𝐵𝑆 → ∀𝑥𝐴 𝐵 ∈ V)
5 mpoexg.1 . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
65mpoexxg 6074 . 2 ((𝐴𝑅 ∧ ∀𝑥𝐴 𝐵 ∈ V) → 𝐹 ∈ V)
74, 6sylan2 282 1 ((𝐴𝑅𝐵𝑆) → 𝐹 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1314  wcel 1463  wral 2391  Vcvv 2658  cmpo 5742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005
This theorem is referenced by:  mpoexga  6076
  Copyright terms: Public domain W3C validator