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Theorem funcestrcsetclem6 16608
 Description: Lemma 6 for funcestrcsetc 16612. (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 16607 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁𝑚 𝑀)))
11103adant3 1074 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑁𝑚 𝑀)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁𝑚 𝑀)))
1211fveq1d 6105 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑁𝑚 𝑀)) → ((𝑋𝐺𝑌)‘𝐻) = (( I ↾ (𝑁𝑚 𝑀))‘𝐻))
13 fvresi 6344 . . 3 (𝐻 ∈ (𝑁𝑚 𝑀) → (( I ↾ (𝑁𝑚 𝑀))‘𝐻) = 𝐻)
14133ad2ant3 1077 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑁𝑚 𝑀)) → (( I ↾ (𝑁𝑚 𝑀))‘𝐻) = 𝐻)
1512, 14eqtrd 2644 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑁𝑚 𝑀)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ↦ cmpt 4643   I cid 4948   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   ↑𝑚 cmap 7744  WUnicwun 9401  Basecbs 15695  SetCatcsetc 16548  ExtStrCatcestrc 16585 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  funcestrcsetclem9  16611  fthestrcsetc  16613  fullestrcsetc  16614
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