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Theorem funcsetcestrclem6 17389
Description: Lemma 6 for funcsetcestrc 17393. (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
funcsetcestrclem6 ((𝜑 ∧ (𝑋𝐶𝑌𝐶) ∧ 𝐻 ∈ (𝑌m 𝑋)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
Distinct variable groups:   𝑥,𝐶   𝑥,𝑋   𝜑,𝑥   𝑦,𝐶,𝑥   𝑦,𝑋   𝑥,𝑌,𝑦   𝜑,𝑦
Allowed substitution hints:   𝑆(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦)

Proof of Theorem funcsetcestrclem6
StepHypRef Expression
1 funcsetcestrc.s . . . . 5 𝑆 = (SetCat‘𝑈)
2 funcsetcestrc.c . . . . 5 𝐶 = (Base‘𝑆)
3 funcsetcestrc.f . . . . 5 (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
4 funcsetcestrc.u . . . . 5 (𝜑𝑈 ∈ WUni)
5 funcsetcestrc.o . . . . 5 (𝜑 → ω ∈ 𝑈)
6 funcsetcestrc.g . . . . 5 (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦m 𝑥))))
71, 2, 3, 4, 5, 6funcsetcestrclem5 17388 . . . 4 ((𝜑 ∧ (𝑋𝐶𝑌𝐶)) → (𝑋𝐺𝑌) = ( I ↾ (𝑌m 𝑋)))
873adant3 1129 . . 3 ((𝜑 ∧ (𝑋𝐶𝑌𝐶) ∧ 𝐻 ∈ (𝑌m 𝑋)) → (𝑋𝐺𝑌) = ( I ↾ (𝑌m 𝑋)))
98fveq1d 6645 . 2 ((𝜑 ∧ (𝑋𝐶𝑌𝐶) ∧ 𝐻 ∈ (𝑌m 𝑋)) → ((𝑋𝐺𝑌)‘𝐻) = (( I ↾ (𝑌m 𝑋))‘𝐻))
10 fvresi 6908 . . 3 (𝐻 ∈ (𝑌m 𝑋) → (( I ↾ (𝑌m 𝑋))‘𝐻) = 𝐻)
11103ad2ant3 1132 . 2 ((𝜑 ∧ (𝑋𝐶𝑌𝐶) ∧ 𝐻 ∈ (𝑌m 𝑋)) → (( I ↾ (𝑌m 𝑋))‘𝐻) = 𝐻)
129, 11eqtrd 2856 1 ((𝜑 ∧ (𝑋𝐶𝑌𝐶) ∧ 𝐻 ∈ (𝑌m 𝑋)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  {csn 4540  cop 4546  cmpt 5119   I cid 5432  cres 5530  cfv 6328  (class class class)co 7130  cmpo 7132  ωcom 7555  m cmap 8381  WUnicwun 10099  ndxcnx 16459  Basecbs 16462  SetCatcsetc 17314
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pr 5303
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135
This theorem is referenced by:  funcsetcestrclem9  17392  fthsetcestrc  17394  fullsetcestrc  17395
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