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Theorem funcestrcsetclem5 18037
Description: Lemma 5 for funcestrcsetc 18042. (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 482 . 2 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))))
3 fveq2 6843 . . . . . 6 (𝑦 = π‘Œ β†’ (Baseβ€˜π‘¦) = (Baseβ€˜π‘Œ))
4 fveq2 6843 . . . . . 6 (π‘₯ = 𝑋 β†’ (Baseβ€˜π‘₯) = (Baseβ€˜π‘‹))
53, 4oveqan12rd 7378 . . . . 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 7370 . . . . 5 (𝑁 ↑m 𝑀) = ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹))
95, 8eqtr4di 2791 . . . 4 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)) = (𝑁 ↑m 𝑀))
109reseq2d 5938 . . 3 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))) = ( I β†Ύ (𝑁 ↑m 𝑀)))
1110adantl 483 . 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 7393 . . 3 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (𝑁 ↑m 𝑀) ∈ V)
1514resiexd 7167 . 2 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ( I β†Ύ (𝑁 ↑m 𝑀)) ∈ V)
162, 11, 12, 13, 15ovmpod 7508 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 397   = wceq 1542   ∈ wcel 2107  Vcvv 3444   ↦ cmpt 5189   I cid 5531   β†Ύ cres 5636  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360   ↑m cmap 8768  WUnicwun 10641  Basecbs 17088  SetCatcsetc 17966  ExtStrCatcestrc 18014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363
This theorem is referenced by:  funcestrcsetclem6  18038  funcestrcsetclem7  18039  funcestrcsetclem8  18040  funcestrcsetclem9  18041
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