Theorem funcsetcestrclem1 18308

Description: Lemma 1 for funcsetcestrc 18318. (Contributed by AV, 27-Mar-2020.)

Hypotheses

Ref	Expression
funcsetcestrc.s	⊢ 𝑆 = (SetCat‘𝑈)
funcsetcestrc.c	⊢ 𝐶 = (Base‘𝑆)
funcsetcestrc.f	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))

Assertion

Ref	Expression
funcsetcestrclem1	⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) = {⟨(Base‘ndx), 𝑋⟩})

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

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

Proof of Theorem funcsetcestrclem1

Step	Hyp	Ref	Expression
1		funcsetcestrc.f	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
2	1	adantr 486	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
3		opeq2 4834	. . . 4 ⊢ (𝑥 = 𝑋 → ⟨(Base‘ndx), 𝑥⟩ = ⟨(Base‘ndx), 𝑋⟩)
4	3	sneqd 4596	. . 3 ⊢ (𝑥 = 𝑋 → {⟨(Base‘ndx), 𝑥⟩} = {⟨(Base‘ndx), 𝑋⟩})
5	4	adantl 487	. 2 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑥 = 𝑋) → {⟨(Base‘ndx), 𝑥⟩} = {⟨(Base‘ndx), 𝑋⟩})
6		simpr 490	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝐶)
7		snex 5397	. . 3 ⊢ {⟨(Base‘ndx), 𝑋⟩} ∈ V
8	7	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → {⟨(Base‘ndx), 𝑋⟩} ∈ V)
9	2, 5, 6, 8	fvmptd 6993	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) = {⟨(Base‘ndx), 𝑋⟩})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451  {csn 4584  ⟨cop 4590   ↦ cmpt 5186  ‘cfv 6531  ndxcnx 17351  Basecbs 17367  SetCatcsetc 18230

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 2733  ax-sep 5249  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  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 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6487  df-fun 6533  df-fv 6539

This theorem is used by:  funcsetcestrclem2  18309  embedsetcestrclem  18311  funcsetcestrclem7  18315  funcsetcestrclem8  18316  funcsetcestrclem9  18317  fullsetcestrc  18320