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Theorem funcsetcestrclem5 18153
Description: Lemma 5 for funcsetcestrc 18158. (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
funcsetcestrclem5 ((𝜑 ∧ (𝑋𝐶𝑌𝐶)) → (𝑋𝐺𝑌) = ( I ↾ (𝑌m 𝑋)))
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)

Proof of Theorem funcsetcestrclem5
StepHypRef Expression
1 funcsetcestrc.g . . 3 (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦m 𝑥))))
21adantr 479 . 2 ((𝜑 ∧ (𝑋𝐶𝑌𝐶)) → 𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦m 𝑥))))
3 oveq12 7428 . . . . 5 ((𝑦 = 𝑌𝑥 = 𝑋) → (𝑦m 𝑥) = (𝑌m 𝑋))
43ancoms 457 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑦m 𝑥) = (𝑌m 𝑋))
54reseq2d 5985 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → ( I ↾ (𝑦m 𝑥)) = ( I ↾ (𝑌m 𝑋)))
65adantl 480 . 2 (((𝜑 ∧ (𝑋𝐶𝑌𝐶)) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ( I ↾ (𝑦m 𝑥)) = ( I ↾ (𝑌m 𝑋)))
7 simprl 769 . 2 ((𝜑 ∧ (𝑋𝐶𝑌𝐶)) → 𝑋𝐶)
8 simprr 771 . 2 ((𝜑 ∧ (𝑋𝐶𝑌𝐶)) → 𝑌𝐶)
9 ovexd 7454 . . 3 ((𝜑 ∧ (𝑋𝐶𝑌𝐶)) → (𝑌m 𝑋) ∈ V)
109resiexd 7228 . 2 ((𝜑 ∧ (𝑋𝐶𝑌𝐶)) → ( I ↾ (𝑌m 𝑋)) ∈ V)
112, 6, 7, 8, 10ovmpod 7573 1 ((𝜑 ∧ (𝑋𝐶𝑌𝐶)) → (𝑋𝐺𝑌) = ( I ↾ (𝑌m 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3461  {csn 4630  cop 4636  cmpt 5232   I cid 5575  cres 5680  cfv 6549  (class class class)co 7419  cmpo 7421  ωcom 7871  m cmap 8845  WUnicwun 10725  ndxcnx 17165  Basecbs 17183  SetCatcsetc 18067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424
This theorem is referenced by:  funcsetcestrclem6  18154  funcsetcestrclem7  18155  funcsetcestrclem8  18156  funcsetcestrclem9  18157
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