Theorem funcringcsetclem1ALTV 49228

Description: Lemma 1 for funcringcsetcALTV 49237. (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 486	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
3		fveq2 6878	. . 3 ⊢ (𝑥 = 𝑋 → (Base‘𝑥) = (Base‘𝑋))
4	3	adantl 487	. 2 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 = 𝑋) → (Base‘𝑥) = (Base‘𝑋))
5		simpr 490	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
6		fvexd 6893	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (Base‘𝑋) ∈ V)
7	2, 4, 5, 6	fvmptd 6994	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ↦ cmpt 5186  ‘cfv 6533  WUnicwun 10709  Basecbs 17301  SetCatcsetc 18164  RingCatALTVcringcALTV 49202

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541

This theorem is used by:  funcringcsetclem2ALTV  49229  funcringcsetclem7ALTV  49234  funcringcsetclem8ALTV  49235  funcringcsetclem9ALTV  49236