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Theorem funcringcsetclem1ALTV 46431
Description: Lemma 1 for funcringcsetcALTV 46440. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
funcringcsetcALTV.r 𝑅 = (RingCatALTVβ€˜π‘ˆ)
funcringcsetcALTV.s 𝑆 = (SetCatβ€˜π‘ˆ)
funcringcsetcALTV.b 𝐡 = (Baseβ€˜π‘…)
funcringcsetcALTV.c 𝐢 = (Baseβ€˜π‘†)
funcringcsetcALTV.u (πœ‘ β†’ π‘ˆ ∈ WUni)
funcringcsetcALTV.f (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)))
Assertion
Ref Expression
funcringcsetclem1ALTV ((πœ‘ ∧ 𝑋 ∈ 𝐡) β†’ (πΉβ€˜π‘‹) = (Baseβ€˜π‘‹))
Distinct variable groups:   π‘₯,𝐡   π‘₯,𝑋   πœ‘,π‘₯
Allowed substitution hints:   𝐢(π‘₯)   𝑅(π‘₯)   𝑆(π‘₯)   π‘ˆ(π‘₯)   𝐹(π‘₯)

Proof of Theorem funcringcsetclem1ALTV
StepHypRef Expression
1 funcringcsetcALTV.f . . 3 (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)))
21adantr 482 . 2 ((πœ‘ ∧ 𝑋 ∈ 𝐡) β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)))
3 fveq2 6847 . . 3 (π‘₯ = 𝑋 β†’ (Baseβ€˜π‘₯) = (Baseβ€˜π‘‹))
43adantl 483 . 2 (((πœ‘ ∧ 𝑋 ∈ 𝐡) ∧ π‘₯ = 𝑋) β†’ (Baseβ€˜π‘₯) = (Baseβ€˜π‘‹))
5 simpr 486 . 2 ((πœ‘ ∧ 𝑋 ∈ 𝐡) β†’ 𝑋 ∈ 𝐡)
6 fvexd 6862 . 2 ((πœ‘ ∧ 𝑋 ∈ 𝐡) β†’ (Baseβ€˜π‘‹) ∈ V)
72, 4, 5, 6fvmptd 6960 1 ((πœ‘ ∧ 𝑋 ∈ 𝐡) β†’ (πΉβ€˜π‘‹) = (Baseβ€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3448   ↦ cmpt 5193  β€˜cfv 6501  WUnicwun 10643  Basecbs 17090  SetCatcsetc 17968  RingCatALTVcringcALTV 46376
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509
This theorem is referenced by:  funcringcsetclem2ALTV  46432  funcringcsetclem7ALTV  46437  funcringcsetclem8ALTV  46438  funcringcsetclem9ALTV  46439
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