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Theorem elixpconstg 39580
 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 elixp2 7954 . . . . . 6 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
21simp2bi 1097 . . . . 5 (𝐹X𝑥𝐴 𝐵𝐹 Fn 𝐴)
31simp3bi 1098 . . . . 5 (𝐹X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
42, 3jca 553 . . . 4 (𝐹X𝑥𝐴 𝐵 → (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
5 ffnfv 6428 . . . 4 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
64, 5sylibr 224 . . 3 (𝐹X𝑥𝐴 𝐵𝐹:𝐴𝐵)
76a1i 11 . 2 (𝐹𝑉 → (𝐹X𝑥𝐴 𝐵𝐹:𝐴𝐵))
8 elex 3243 . . . . . 6 (𝐹𝑉𝐹 ∈ V)
98adantr 480 . . . . 5 ((𝐹𝑉𝐹:𝐴𝐵) → 𝐹 ∈ V)
10 ffn 6083 . . . . . 6 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
1110adantl 481 . . . . 5 ((𝐹𝑉𝐹:𝐴𝐵) → 𝐹 Fn 𝐴)
125simprbi 479 . . . . . 6 (𝐹:𝐴𝐵 → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
1312adantl 481 . . . . 5 ((𝐹𝑉𝐹:𝐴𝐵) → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
149, 11, 133jca 1261 . . . 4 ((𝐹𝑉𝐹:𝐴𝐵) → (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
1514, 1sylibr 224 . . 3 ((𝐹𝑉𝐹:𝐴𝐵) → 𝐹X𝑥𝐴 𝐵)
1615ex 449 . 2 (𝐹𝑉 → (𝐹:𝐴𝐵𝐹X𝑥𝐴 𝐵))
177, 16impbid 202 1 (𝐹𝑉 → (𝐹X𝑥𝐴 𝐵𝐹:𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   ∈ wcel 2030  ∀wral 2941  Vcvv 3231   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  Xcixp 7950 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ixp 7951 This theorem is referenced by:  iinhoiicclem  41208
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