Theorem funresdfunsn 7223

Description: Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in the function itself. (Contributed by AV, 2-Dec-2018.)

Assertion

Ref	Expression
funresdfunsn	⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}) = 𝐹)

Proof of Theorem funresdfunsn

Step	Hyp	Ref	Expression
1		funrel 6595	. . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹)
2		resdmdfsn 6060	. . . . 5 ⊢ (Rel 𝐹 → (𝐹 ↾ (V ∖ {𝑋})) = (𝐹 ↾ (dom 𝐹 ∖ {𝑋})))
3	1, 2	syl 17	. . . 4 ⊢ (Fun 𝐹 → (𝐹 ↾ (V ∖ {𝑋})) = (𝐹 ↾ (dom 𝐹 ∖ {𝑋})))
4	3	adantr 480	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ (V ∖ {𝑋})) = (𝐹 ↾ (dom 𝐹 ∖ {𝑋})))
5	4	uneq1d 4190	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}) = ((𝐹 ↾ (dom 𝐹 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}))
6		funfn 6608	. . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
7		fnsnsplit 7218	. . 3 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝑋 ∈ dom 𝐹) → 𝐹 = ((𝐹 ↾ (dom 𝐹 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}))
8	6, 7	sylanb 580	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → 𝐹 = ((𝐹 ↾ (dom 𝐹 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}))
9	5, 8	eqtr4d 2783	1 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∖ cdif 3973   ∪ cun 3974  {csn 4648  ⟨cop 4654  dom cdm 5700   ↾ cres 5702  Rel wrel 5705  Fun wfun 6567   Fn wfn 6568  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581

This theorem is referenced by:  setsidvald  17246  setsidvaldOLD  17247