Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elixpconstg Structured version   Visualization version   GIF version

Theorem elixpconstg 41217
Description: Membership in an infinite Cartesian product of a constant 𝐵. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
Assertion
Ref Expression
elixpconstg (𝐹𝑉 → (𝐹X𝑥𝐴 𝐵𝐹:𝐴𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem elixpconstg
StepHypRef Expression
1 ixpfn 8459 . . 3 (𝐹X𝑥𝐴 𝐵𝐹 Fn 𝐴)
2 elixp2 8457 . . . 4 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
32simp3bi 1141 . . 3 (𝐹X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
4 ffnfv 6877 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
51, 3, 4sylanbrc 583 . 2 (𝐹X𝑥𝐴 𝐵𝐹:𝐴𝐵)
6 elex 3517 . . . . 5 (𝐹𝑉𝐹 ∈ V)
76adantr 481 . . . 4 ((𝐹𝑉𝐹:𝐴𝐵) → 𝐹 ∈ V)
8 ffn 6510 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
98adantl 482 . . . 4 ((𝐹𝑉𝐹:𝐴𝐵) → 𝐹 Fn 𝐴)
104simprbi 497 . . . . 5 (𝐹:𝐴𝐵 → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
1110adantl 482 . . . 4 ((𝐹𝑉𝐹:𝐴𝐵) → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
127, 9, 11, 2syl3anbrc 1337 . . 3 ((𝐹𝑉𝐹:𝐴𝐵) → 𝐹X𝑥𝐴 𝐵)
1312ex 413 . 2 (𝐹𝑉 → (𝐹:𝐴𝐵𝐹X𝑥𝐴 𝐵))
145, 13impbid2 227 1 (𝐹𝑉 → (𝐹X𝑥𝐴 𝐵𝐹:𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2106  wral 3142  Vcvv 3499   Fn wfn 6346  wf 6347  cfv 6351  Xcixp 8453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fv 6359  df-ixp 8454
This theorem is referenced by:  iinhoiicclem  42818
  Copyright terms: Public domain W3C validator