Theorem funcringcsetclem1ALTV 48229

Description: Lemma 1 for funcringcsetcALTV 48238. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)

Hypotheses

Ref	Expression
funcringcsetcALTV.r	⊢ 𝑅 = (RingCatALTV‘𝑈)
funcringcsetcALTV.s	⊢ 𝑆 = (SetCat‘𝑈)
funcringcsetcALTV.b	⊢ 𝐵 = (Base‘𝑅)
funcringcsetcALTV.c	⊢ 𝐶 = (Base‘𝑆)
funcringcsetcALTV.u	⊢ (𝜑 → 𝑈 ∈ WUni)
funcringcsetcALTV.f	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))

Assertion

Ref	Expression
funcringcsetclem1ALTV	⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋))

Distinct variable groups:   𝑥,𝐵   𝑥,𝑋   𝜑,𝑥

Allowed substitution hints:   𝐶(𝑥)   𝑅(𝑥)   𝑆(𝑥)   𝑈(𝑥)   𝐹(𝑥)

Proof of Theorem funcringcsetclem1ALTV

Step	Hyp	Ref	Expression
1		funcringcsetcALTV.f	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
2	1	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
3		fveq2 6906	. . 3 ⊢ (𝑥 = 𝑋 → (Base‘𝑥) = (Base‘𝑋))
4	3	adantl 481	. 2 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 = 𝑋) → (Base‘𝑥) = (Base‘𝑋))
5		simpr 484	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
6		fvexd 6921	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (Base‘𝑋) ∈ V)
7	2, 4, 5, 6	fvmptd 7023	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3480   ↦ cmpt 5225  ‘cfv 6561  WUnicwun 10740  Basecbs 17247  SetCatcsetc 18120  RingCatALTVcringcALTV 48203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569

This theorem is referenced by:  funcringcsetclem2ALTV  48230  funcringcsetclem7ALTV  48235  funcringcsetclem8ALTV  48236  funcringcsetclem9ALTV  48237