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Theorem funcestrcsetclem5 17096
 Description: Lemma 5 for funcestrcsetc 17101. (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
funcestrcsetclem5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁𝑚 𝑀)))
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)

Proof of Theorem funcestrcsetclem5
StepHypRef Expression
1 funcestrcsetc.g . . 3 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))
21adantr 473 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))
3 fveq2 6410 . . . . . 6 (𝑦 = 𝑌 → (Base‘𝑦) = (Base‘𝑌))
4 fveq2 6410 . . . . . 6 (𝑥 = 𝑋 → (Base‘𝑥) = (Base‘𝑋))
53, 4oveqan12rd 6897 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
6 funcestrcsetc.n . . . . . 6 𝑁 = (Base‘𝑌)
7 funcestrcsetc.m . . . . . 6 𝑀 = (Base‘𝑋)
86, 7oveq12i 6889 . . . . 5 (𝑁𝑚 𝑀) = ((Base‘𝑌) ↑𝑚 (Base‘𝑋))
95, 8syl6eqr 2850 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = (𝑁𝑚 𝑀))
109reseq2d 5599 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = ( I ↾ (𝑁𝑚 𝑀)))
1110adantl 474 . 2 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = ( I ↾ (𝑁𝑚 𝑀)))
12 simprl 788 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
13 simprr 790 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
14 ovexd 6911 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑁𝑚 𝑀) ∈ V)
1514resiexd 6708 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ( I ↾ (𝑁𝑚 𝑀)) ∈ V)
162, 11, 12, 13, 15ovmpt2d 7021 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁𝑚 𝑀)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157  Vcvv 3384   ↦ cmpt 4921   I cid 5218   ↾ cres 5313  ‘cfv 6100  (class class class)co 6877   ↦ cmpt2 6879   ↑𝑚 cmap 8094  WUnicwun 9809  Basecbs 16181  SetCatcsetc 17036  ExtStrCatcestrc 17073 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 2776  ax-rep 4963  ax-sep 4974  ax-nul 4982  ax-pr 5096 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 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3386  df-sbc 3633  df-csb 3728  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4628  df-iun 4711  df-br 4843  df-opab 4905  df-mpt 4922  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6063  df-fun 6102  df-fn 6103  df-f 6104  df-f1 6105  df-fo 6106  df-f1o 6107  df-fv 6108  df-ov 6880  df-oprab 6881  df-mpt2 6882 This theorem is referenced by:  funcestrcsetclem6  17097  funcestrcsetclem7  17098  funcestrcsetclem8  17099  funcestrcsetclem9  17100
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