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Theorem funcestrcsetclem5 18126
Description: Lemma 5 for funcestrcsetc 18131. (Contributed by AV, 23-Mar-2020.)
Hypotheses
Ref Expression
funcestrcsetc.e 𝐸 = (ExtStrCatβ€˜π‘ˆ)
funcestrcsetc.s 𝑆 = (SetCatβ€˜π‘ˆ)
funcestrcsetc.b 𝐡 = (Baseβ€˜πΈ)
funcestrcsetc.c 𝐢 = (Baseβ€˜π‘†)
funcestrcsetc.u (πœ‘ β†’ π‘ˆ ∈ WUni)
funcestrcsetc.f (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)))
funcestrcsetc.g (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))))
funcestrcsetc.m 𝑀 = (Baseβ€˜π‘‹)
funcestrcsetc.n 𝑁 = (Baseβ€˜π‘Œ)
Assertion
Ref Expression
funcestrcsetclem5 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (π‘‹πΊπ‘Œ) = ( I β†Ύ (𝑁 ↑m 𝑀)))
Distinct variable groups:   π‘₯,𝐡   π‘₯,𝑋   πœ‘,π‘₯   π‘₯,𝐢   𝑦,𝐡,π‘₯   𝑦,𝑋   πœ‘,𝑦   π‘₯,𝑀,𝑦   π‘₯,𝑁,𝑦   π‘₯,π‘Œ,𝑦
Allowed substitution hints:   𝐢(𝑦)   𝑆(π‘₯,𝑦)   π‘ˆ(π‘₯,𝑦)   𝐸(π‘₯,𝑦)   𝐹(π‘₯,𝑦)   𝐺(π‘₯,𝑦)

Proof of Theorem funcestrcsetclem5
StepHypRef Expression
1 funcestrcsetc.g . . 3 (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))))
21adantr 480 . 2 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))))
3 fveq2 6891 . . . . . 6 (𝑦 = π‘Œ β†’ (Baseβ€˜π‘¦) = (Baseβ€˜π‘Œ))
4 fveq2 6891 . . . . . 6 (π‘₯ = 𝑋 β†’ (Baseβ€˜π‘₯) = (Baseβ€˜π‘‹))
53, 4oveqan12rd 7434 . . . . 5 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)) = ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)))
6 funcestrcsetc.n . . . . . 6 𝑁 = (Baseβ€˜π‘Œ)
7 funcestrcsetc.m . . . . . 6 𝑀 = (Baseβ€˜π‘‹)
86, 7oveq12i 7426 . . . . 5 (𝑁 ↑m 𝑀) = ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹))
95, 8eqtr4di 2785 . . . 4 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)) = (𝑁 ↑m 𝑀))
109reseq2d 5979 . . 3 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))) = ( I β†Ύ (𝑁 ↑m 𝑀)))
1110adantl 481 . 2 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) ∧ (π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ)) β†’ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))) = ( I β†Ύ (𝑁 ↑m 𝑀)))
12 simprl 770 . 2 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ 𝑋 ∈ 𝐡)
13 simprr 772 . 2 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ π‘Œ ∈ 𝐡)
14 ovexd 7449 . . 3 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (𝑁 ↑m 𝑀) ∈ V)
1514resiexd 7222 . 2 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ( I β†Ύ (𝑁 ↑m 𝑀)) ∈ V)
162, 11, 12, 13, 15ovmpod 7567 1 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (π‘‹πΊπ‘Œ) = ( I β†Ύ (𝑁 ↑m 𝑀)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  Vcvv 3469   ↦ cmpt 5225   I cid 5569   β†Ύ cres 5674  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416   ↑m cmap 8836  WUnicwun 10715  Basecbs 17171  SetCatcsetc 18055  ExtStrCatcestrc 18103
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 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423
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 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419
This theorem is referenced by:  funcestrcsetclem6  18127  funcestrcsetclem7  18128  funcestrcsetclem8  18129  funcestrcsetclem9  18130
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