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Theorem funcringcsetcALTV2lem6 42840
Description: Lemma 6 for funcringcsetcALTV2 42844. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
funcringcsetcALTV2.r 𝑅 = (RingCat‘𝑈)
funcringcsetcALTV2.s 𝑆 = (SetCat‘𝑈)
funcringcsetcALTV2.b 𝐵 = (Base‘𝑅)
funcringcsetcALTV2.c 𝐶 = (Base‘𝑆)
funcringcsetcALTV2.u (𝜑𝑈 ∈ WUni)
funcringcsetcALTV2.f (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
funcringcsetcALTV2.g (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
Assertion
Ref Expression
funcringcsetcALTV2lem6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑋 RingHom 𝑌)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
Distinct variable groups:   𝑥,𝐵   𝑥,𝑋   𝜑,𝑥   𝑥,𝐶   𝑦,𝐵,𝑥   𝑦,𝑋   𝑥,𝑌,𝑦   𝜑,𝑦
Allowed substitution hints:   𝐶(𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦)

Proof of Theorem funcringcsetcALTV2lem6
StepHypRef Expression
1 funcringcsetcALTV2.r . . . . 5 𝑅 = (RingCat‘𝑈)
2 funcringcsetcALTV2.s . . . . 5 𝑆 = (SetCat‘𝑈)
3 funcringcsetcALTV2.b . . . . 5 𝐵 = (Base‘𝑅)
4 funcringcsetcALTV2.c . . . . 5 𝐶 = (Base‘𝑆)
5 funcringcsetcALTV2.u . . . . 5 (𝜑𝑈 ∈ WUni)
6 funcringcsetcALTV2.f . . . . 5 (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
7 funcringcsetcALTV2.g . . . . 5 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
81, 2, 3, 4, 5, 6, 7funcringcsetcALTV2lem5 42839 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑋 RingHom 𝑌)))
983adant3 1163 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑋 RingHom 𝑌)) → (𝑋𝐺𝑌) = ( I ↾ (𝑋 RingHom 𝑌)))
109fveq1d 6413 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑋 RingHom 𝑌)) → ((𝑋𝐺𝑌)‘𝐻) = (( I ↾ (𝑋 RingHom 𝑌))‘𝐻))
11 fvresi 6668 . . 3 (𝐻 ∈ (𝑋 RingHom 𝑌) → (( I ↾ (𝑋 RingHom 𝑌))‘𝐻) = 𝐻)
12113ad2ant3 1166 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑋 RingHom 𝑌)) → (( I ↾ (𝑋 RingHom 𝑌))‘𝐻) = 𝐻)
1310, 12eqtrd 2833 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑋 RingHom 𝑌)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  cmpt 4922   I cid 5219  cres 5314  cfv 6101  (class class class)co 6878  cmpt2 6880  WUnicwun 9810  Basecbs 16184  SetCatcsetc 17039   RingHom crh 19030  RingCatcringc 42802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883
This theorem is referenced by:  funcringcsetcALTV2lem9  42843
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