Theorem funcringcsetclem1ALTV 48555

Description: Lemma 1 for funcringcsetcALTV 48564. (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 6834	. . 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 6849	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (Base‘𝑋) ∈ V)
7	2, 4, 5, 6	fvmptd 6948	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ↦ cmpt 5179  ‘cfv 6492  WUnicwun 10611  Basecbs 17136  SetCatcsetc 17999  RingCatALTVcringcALTV 48529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500

This theorem is referenced by:  funcringcsetclem2ALTV  48556  funcringcsetclem7ALTV  48561  funcringcsetclem8ALTV  48562  funcringcsetclem9ALTV  48563