Theorem funcsetcestrclem1 18111

Description: Lemma 1 for funcsetcestrc 18121. (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 480	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
3		opeq2 4874	. . . 4 ⊢ (𝑥 = 𝑋 → ⟨(Base‘ndx), 𝑥⟩ = ⟨(Base‘ndx), 𝑋⟩)
4	3	sneqd 4640	. . 3 ⊢ (𝑥 = 𝑋 → {⟨(Base‘ndx), 𝑥⟩} = {⟨(Base‘ndx), 𝑋⟩})
5	4	adantl 481	. 2 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑥 = 𝑋) → {⟨(Base‘ndx), 𝑥⟩} = {⟨(Base‘ndx), 𝑋⟩})
6		simpr 484	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝐶)
7		snex 5431	. . 3 ⊢ {⟨(Base‘ndx), 𝑋⟩} ∈ V
8	7	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → {⟨(Base‘ndx), 𝑋⟩} ∈ V)
9	2, 5, 6, 8	fvmptd 7005	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) = {⟨(Base‘ndx), 𝑋⟩})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  Vcvv 3473  {csn 4628  ⟨cop 4634   ↦ cmpt 5231  ‘cfv 6543  ndxcnx 17131  Basecbs 17149  SetCatcsetc 18030

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551

This theorem is referenced by:  funcsetcestrclem2  18112  embedsetcestrclem  18114  funcsetcestrclem7  18118  funcsetcestrclem8  18119  funcsetcestrclem9  18120  fullsetcestrc  18123