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Theorem ialgrlemconst 10650
Description: Lemma for ialgr0 10651. Closure of a constant function, in a form suitable for theorems such as iseq1 9603 or iseqfcl 9605. (Contributed by Jim Kingdon, 22-Jul-2021.)
Hypotheses
Ref Expression
ialgrlemconst.z 𝑍 = (ℤ𝑀)
ialgrlemconst.a (𝜑𝐴𝑆)
Assertion
Ref Expression
ialgrlemconst ((𝜑𝑥 ∈ (ℤ𝑀)) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆)

Proof of Theorem ialgrlemconst
StepHypRef Expression
1 ialgrlemconst.a . . 3 (𝜑𝐴𝑆)
2 ialgrlemconst.z . . . . 5 𝑍 = (ℤ𝑀)
32eleq2i 2149 . . . 4 (𝑥𝑍𝑥 ∈ (ℤ𝑀))
43biimpri 131 . . 3 (𝑥 ∈ (ℤ𝑀) → 𝑥𝑍)
5 fvconst2g 5428 . . 3 ((𝐴𝑆𝑥𝑍) → ((𝑍 × {𝐴})‘𝑥) = 𝐴)
61, 4, 5syl2an 283 . 2 ((𝜑𝑥 ∈ (ℤ𝑀)) → ((𝑍 × {𝐴})‘𝑥) = 𝐴)
71adantr 270 . 2 ((𝜑𝑥 ∈ (ℤ𝑀)) → 𝐴𝑆)
86, 7eqeltrd 2159 1 ((𝜑𝑥 ∈ (ℤ𝑀)) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  {csn 3416   × cxp 4389  cfv 4952  cuz 8770
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-fv 4960
This theorem is referenced by:  ialgr0  10651  ialgrf  10652  ialgrp1  10653
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