Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  funcsetcestrclem4 Structured version   Visualization version   GIF version

Theorem funcsetcestrclem4 17403
 Description: Lemma 4 for funcsetcestrc 17409. (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 2798 . . 3 (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦m 𝑥))) = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦m 𝑥)))
2 ovex 7169 . . . 4 (𝑦m 𝑥) ∈ V
3 resiexg 7604 . . . 4 ((𝑦m 𝑥) ∈ V → ( I ↾ (𝑦m 𝑥)) ∈ V)
42, 3ax-mp 5 . . 3 ( I ↾ (𝑦m 𝑥)) ∈ V
51, 4fnmpoi 7753 . 2 (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦m 𝑥))) Fn (𝐶 × 𝐶)
6 funcsetcestrc.g . . 3 (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦m 𝑥))))
76fneq1d 6417 . 2 (𝜑 → (𝐺 Fn (𝐶 × 𝐶) ↔ (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦m 𝑥))) Fn (𝐶 × 𝐶)))
85, 7mpbiri 261 1 (𝜑𝐺 Fn (𝐶 × 𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3441  {csn 4525  ⟨cop 4531   ↦ cmpt 5111   I cid 5425   × cxp 5518   ↾ cres 5522   Fn wfn 6320  ‘cfv 6325  (class class class)co 7136   ∈ cmpo 7138  ωcom 7563   ↑m cmap 8392  WUnicwun 10114  ndxcnx 16475  Basecbs 16478  SetCatcsetc 17330 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-1st 7674  df-2nd 7675 This theorem is referenced by:  funcsetcestrc  17409
 Copyright terms: Public domain W3C validator