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Theorem funcsetcestrclem4 17396
Description: Lemma 4 for funcsetcestrc 17402. (Contributed by AV, 27-Mar-2020.)
Hypotheses
Ref Expression
funcsetcestrc.s 𝑆 = (SetCat‘𝑈)
funcsetcestrc.c 𝐶 = (Base‘𝑆)
funcsetcestrc.f (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
funcsetcestrc.u (𝜑𝑈 ∈ WUni)
funcsetcestrc.o (𝜑 → ω ∈ 𝑈)
funcsetcestrc.g (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦m 𝑥))))
Assertion
Ref Expression
funcsetcestrclem4 (𝜑𝐺 Fn (𝐶 × 𝐶))
Distinct variable groups:   𝑥,𝐶   𝜑,𝑥   𝑦,𝐶,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝑆(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem funcsetcestrclem4
StepHypRef Expression
1 eqid 2818 . . 3 (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦m 𝑥))) = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦m 𝑥)))
2 ovex 7178 . . . 4 (𝑦m 𝑥) ∈ V
3 resiexg 7608 . . . 4 ((𝑦m 𝑥) ∈ V → ( I ↾ (𝑦m 𝑥)) ∈ V)
42, 3ax-mp 5 . . 3 ( I ↾ (𝑦m 𝑥)) ∈ V
51, 4fnmpoi 7757 . 2 (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦m 𝑥))) Fn (𝐶 × 𝐶)
6 funcsetcestrc.g . . 3 (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦m 𝑥))))
76fneq1d 6439 . 2 (𝜑 → (𝐺 Fn (𝐶 × 𝐶) ↔ (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦m 𝑥))) Fn (𝐶 × 𝐶)))
85, 7mpbiri 259 1 (𝜑𝐺 Fn (𝐶 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  Vcvv 3492  {csn 4557  cop 4563  cmpt 5137   I cid 5452   × cxp 5546  cres 5550   Fn wfn 6343  cfv 6348  (class class class)co 7145  cmpo 7147  ωcom 7569  m cmap 8395  WUnicwun 10110  ndxcnx 16468  Basecbs 16471  SetCatcsetc 17323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679
This theorem is referenced by:  funcsetcestrc  17402
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