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Theorem elixpconst 8445
Description: Membership in an infinite Cartesian product of a constant 𝐵. (Contributed by NM, 12-Apr-2008.)
Hypothesis
Ref Expression
elixp.1 𝐹 ∈ V
Assertion
Ref Expression
elixpconst (𝐹X𝑥𝐴 𝐵𝐹:𝐴𝐵)
Distinct variable groups:   𝑥,𝐹   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elixpconst
StepHypRef Expression
1 elixp.1 . . 3 𝐹 ∈ V
21elixp 8444 . 2 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
3 ffnfv 6856 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
42, 3bitr4i 280 1 (𝐹X𝑥𝐴 𝐵𝐹:𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398  wcel 2114  wral 3125  Vcvv 3473   Fn wfn 6324  wf 6325  cfv 6329  Xcixp 8437
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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5177  ax-nul 5184  ax-pr 5304
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3752  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-nul 4268  df-if 4442  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4813  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5434  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-fv 6337  df-ixp 8438
This theorem is referenced by:  ixpconstg  8446  sscpwex  17061  psrbaglefi  20125
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