MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funcestrcsetclem6 Structured version   Visualization version   GIF version

Theorem funcestrcsetclem6 18106
Description: Lemma 6 for funcestrcsetc 18110. (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 18105 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁m 𝑀)))
11103adant3 1139 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑁m 𝑀)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁m 𝑀)))
1211fveq1d 6832 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑁m 𝑀)) → ((𝑋𝐺𝑌)‘𝐻) = (( I ↾ (𝑁m 𝑀))‘𝐻))
13 fvresi 7120 . . 3 (𝐻 ∈ (𝑁m 𝑀) → (( I ↾ (𝑁m 𝑀))‘𝐻) = 𝐻)
14133ad2ant3 1142 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑁m 𝑀)) → (( I ↾ (𝑁m 𝑀))‘𝐻) = 𝐻)
1512, 14eqtrd 2776 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑁m 𝑀)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1093   = wceq 1548  wcel 2121  cmpt 5155   I cid 5514  cres 5622  cfv 6488  (class class class)co 7359  cmpo 7361  m cmap 8767  WUnicwun 10619  Basecbs 17174  SetCatcsetc 18037  ExtStrCatcestrc 18083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pr 5364
This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7362  df-oprab 7363  df-mpo 7364
This theorem is referenced by:  funcestrcsetclem9  18109  fthestrcsetc  18111  fullestrcsetc  18112
  Copyright terms: Public domain W3C validator