Theorem setsabs 17108

Description: Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.)

Assertion

Ref	Expression
setsabs	⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑆 sSet ⟨𝐴, 𝐵⟩) sSet ⟨𝐴, 𝐶⟩) = (𝑆 sSet ⟨𝐴, 𝐶⟩))

Proof of Theorem setsabs

Step	Hyp	Ref	Expression
1		setsres 17107	. . . 4 ⊢ (𝑆 ∈ 𝑉 → ((𝑆 sSet ⟨𝐴, 𝐵⟩) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴})))
2	1	adantr 480	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑆 sSet ⟨𝐴, 𝐵⟩) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴})))
3	2	uneq1d 4120	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (((𝑆 sSet ⟨𝐴, 𝐵⟩) ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐶⟩}) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐶⟩}))
4		ovexd 7388	. . 3 ⊢ (𝑆 ∈ 𝑉 → (𝑆 sSet ⟨𝐴, 𝐵⟩) ∈ V)
5		setsval 17096	. . 3 ⊢ (((𝑆 sSet ⟨𝐴, 𝐵⟩) ∈ V ∧ 𝐶 ∈ 𝑊) → ((𝑆 sSet ⟨𝐴, 𝐵⟩) sSet ⟨𝐴, 𝐶⟩) = (((𝑆 sSet ⟨𝐴, 𝐵⟩) ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐶⟩}))
6	4, 5	sylan 580	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑆 sSet ⟨𝐴, 𝐵⟩) sSet ⟨𝐴, 𝐶⟩) = (((𝑆 sSet ⟨𝐴, 𝐵⟩) ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐶⟩}))
7		setsval 17096	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑆 sSet ⟨𝐴, 𝐶⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐶⟩}))
8	3, 6, 7	3eqtr4d 2774	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑆 sSet ⟨𝐴, 𝐵⟩) sSet ⟨𝐴, 𝐶⟩) = (𝑆 sSet ⟨𝐴, 𝐶⟩))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3438   ∖ cdif 3902   ∪ cun 3903  {csn 4579  ⟨cop 4585   ↾ cres 5625  (class class class)co 7353   sSet csts 17092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-res 5635  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-sets 17093

This theorem is referenced by:  ressress  17176  rescabs  17758  opprabs  33429