Theorem setsabs 16629

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 16628	. . . 4 ⊢ (𝑆 ∈ 𝑉 → ((𝑆 sSet ⟨𝐴, 𝐵⟩) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴})))
2	1	adantr 484	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑆 sSet ⟨𝐴, 𝐵⟩) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴})))
3	2	uneq1d 4052	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (((𝑆 sSet ⟨𝐴, 𝐵⟩) ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐶⟩}) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐶⟩}))
4		ovexd 7205	. . 3 ⊢ (𝑆 ∈ 𝑉 → (𝑆 sSet ⟨𝐴, 𝐵⟩) ∈ V)
5		setsval 16616	. . 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 583	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑆 sSet ⟨𝐴, 𝐵⟩) sSet ⟨𝐴, 𝐶⟩) = (((𝑆 sSet ⟨𝐴, 𝐵⟩) ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐶⟩}))
7		setsval 16616	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑆 sSet ⟨𝐴, 𝐶⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐶⟩}))
8	3, 6, 7	3eqtr4d 2783	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑆 sSet ⟨𝐴, 𝐵⟩) sSet ⟨𝐴, 𝐶⟩) = (𝑆 sSet ⟨𝐴, 𝐶⟩))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1542   ∈ wcel 2114  Vcvv 3398   ∖ cdif 3840   ∪ cun 3841  {csn 4516  ⟨cop 4522   ↾ cres 5527  (class class class)co 7170   sSet csts 16584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-res 5537  df-iota 6297  df-fun 6341  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-sets 16593

This theorem is referenced by:  ressress  16665  rescabs  17208