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Theorem funcestrcsetclem6 18136
Description: Lemma 6 for funcestrcsetc 18140. (Contributed by AV, 23-Mar-2020.)
Hypotheses
Ref Expression
funcestrcsetc.e 𝐸 = (ExtStrCat‘𝑈)
funcestrcsetc.s 𝑆 = (SetCat‘𝑈)
funcestrcsetc.b 𝐵 = (Base‘𝐸)
funcestrcsetc.c 𝐶 = (Base‘𝑆)
funcestrcsetc.u (𝜑𝑈 ∈ WUni)
funcestrcsetc.f (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
funcestrcsetc.g (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
funcestrcsetc.m 𝑀 = (Base‘𝑋)
funcestrcsetc.n 𝑁 = (Base‘𝑌)
Assertion
Ref Expression
funcestrcsetclem6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑁m 𝑀)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
Distinct variable groups:   𝑥,𝐵   𝑥,𝑋   𝜑,𝑥   𝑥,𝐶   𝑦,𝐵,𝑥   𝑦,𝑋   𝜑,𝑦   𝑥,𝑀,𝑦   𝑥,𝑁,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐶(𝑦)   𝑆(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦)

Proof of Theorem funcestrcsetclem6
StepHypRef Expression
1 funcestrcsetc.e . . . . 5 𝐸 = (ExtStrCat‘𝑈)
2 funcestrcsetc.s . . . . 5 𝑆 = (SetCat‘𝑈)
3 funcestrcsetc.b . . . . 5 𝐵 = (Base‘𝐸)
4 funcestrcsetc.c . . . . 5 𝐶 = (Base‘𝑆)
5 funcestrcsetc.u . . . . 5 (𝜑𝑈 ∈ WUni)
6 funcestrcsetc.f . . . . 5 (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
7 funcestrcsetc.g . . . . 5 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
8 funcestrcsetc.m . . . . 5 𝑀 = (Base‘𝑋)
9 funcestrcsetc.n . . . . 5 𝑁 = (Base‘𝑌)
101, 2, 3, 4, 5, 6, 7, 8, 9funcestrcsetclem5 18135 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁m 𝑀)))
11103adant3 1130 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑁m 𝑀)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁m 𝑀)))
1211fveq1d 6899 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑁m 𝑀)) → ((𝑋𝐺𝑌)‘𝐻) = (( I ↾ (𝑁m 𝑀))‘𝐻))
13 fvresi 7182 . . 3 (𝐻 ∈ (𝑁m 𝑀) → (( I ↾ (𝑁m 𝑀))‘𝐻) = 𝐻)
14133ad2ant3 1133 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑁m 𝑀)) → (( I ↾ (𝑁m 𝑀))‘𝐻) = 𝐻)
1512, 14eqtrd 2768 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑁m 𝑀)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  cmpt 5231   I cid 5575  cres 5680  cfv 6548  (class class class)co 7420  cmpo 7422  m cmap 8845  WUnicwun 10724  Basecbs 17180  SetCatcsetc 18064  ExtStrCatcestrc 18112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  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 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425
This theorem is referenced by:  funcestrcsetclem9  18139  fthestrcsetc  18141  fullestrcsetc  18142
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