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Theorem funcestrcsetclem4 18095
Description: Lemma 4 for funcestrcsetc 18101. (Contributed by AV, 22-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‘₯)))))
Assertion
Ref Expression
funcestrcsetclem4 (πœ‘ β†’ 𝐺 Fn (𝐡 Γ— 𝐡))
Distinct variable groups:   π‘₯,𝐡   πœ‘,π‘₯   π‘₯,𝐢   𝑦,𝐡,π‘₯
Allowed substitution hints:   πœ‘(𝑦)   𝐢(𝑦)   𝑆(π‘₯,𝑦)   π‘ˆ(π‘₯,𝑦)   𝐸(π‘₯,𝑦)   𝐹(π‘₯,𝑦)   𝐺(π‘₯,𝑦)

Proof of Theorem funcestrcsetclem4
StepHypRef Expression
1 eqid 2733 . . 3 (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))) = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))))
2 ovex 7442 . . . 4 ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)) ∈ V
3 resiexg 7905 . . . 4 (((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)) ∈ V β†’ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))) ∈ V)
42, 3ax-mp 5 . . 3 ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))) ∈ V
51, 4fnmpoi 8056 . 2 (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))) Fn (𝐡 Γ— 𝐡)
6 funcestrcsetc.g . . 3 (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))))
76fneq1d 6643 . 2 (πœ‘ β†’ (𝐺 Fn (𝐡 Γ— 𝐡) ↔ (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))) Fn (𝐡 Γ— 𝐡)))
85, 7mpbiri 258 1 (πœ‘ β†’ 𝐺 Fn (𝐡 Γ— 𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ↦ cmpt 5232   I cid 5574   Γ— cxp 5675   β†Ύ cres 5679   Fn wfn 6539  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411   ↑m cmap 8820  WUnicwun 10695  Basecbs 17144  SetCatcsetc 18025  ExtStrCatcestrc 18073
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-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976
This theorem is referenced by:  funcestrcsetc  18101
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