Theorem funcringcsetcALTV2lem1 47464

Description: Lemma 1 for funcringcsetcALTV2 47473. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)

Hypotheses

Ref	Expression
funcringcsetcALTV2.r	⊢ 𝑅 = (RingCat‘𝑈)
funcringcsetcALTV2.s	⊢ 𝑆 = (SetCat‘𝑈)
funcringcsetcALTV2.b	⊢ 𝐵 = (Base‘𝑅)
funcringcsetcALTV2.c	⊢ 𝐶 = (Base‘𝑆)
funcringcsetcALTV2.u	⊢ (𝜑 → 𝑈 ∈ WUni)
funcringcsetcALTV2.f	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))

Assertion

Ref	Expression
funcringcsetcALTV2lem1	⊢ ((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 funcringcsetcALTV2lem1

Step	Hyp	Ref	Expression
1		funcringcsetcALTV2.f	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
2	1	adantr 479	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
3		fveq2 6894	. . 3 ⊢ (𝑥 = 𝑋 → (Base‘𝑥) = (Base‘𝑋))
4	3	adantl 480	. 2 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 = 𝑋) → (Base‘𝑥) = (Base‘𝑋))
5		simpr 483	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
6		fvexd 6909	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (Base‘𝑋) ∈ V)
7	2, 4, 5, 6	fvmptd 7009	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3463   ↦ cmpt 5231  ‘cfv 6547  WUnicwun 10723  Basecbs 17179  SetCatcsetc 18063  RingCatcringc 20582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5299  ax-nul 5306  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-ss 3962  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6499  df-fun 6549  df-fv 6555

This theorem is referenced by:  funcringcsetcALTV2lem2  47465  funcringcsetcALTV2lem7  47470  funcringcsetcALTV2lem8  47471  funcringcsetcALTV2lem9  47472