Theorem funresdfunsnss 5815

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 a subset of the function itself. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 21-Jan-2023.)

Assertion

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

Proof of Theorem funresdfunsnss

Step	Hyp	Ref	Expression
1		funrel 5311	. . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹)
2		resdmdfsn 5024	. . . . 5 ⊢ (Rel 𝐹 → (𝐹 ↾ (V ∖ {𝑋})) = (𝐹 ↾ (dom 𝐹 ∖ {𝑋})))
3	1, 2	syl 14	. . . 4 ⊢ (Fun 𝐹 → (𝐹 ↾ (V ∖ {𝑋})) = (𝐹 ↾ (dom 𝐹 ∖ {𝑋})))
4	3	adantr 276	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ (V ∖ {𝑋})) = (𝐹 ↾ (dom 𝐹 ∖ {𝑋})))
5	4	uneq1d 3337	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}) = ((𝐹 ↾ (dom 𝐹 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}))
6		funfn 5324	. . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
7		fnsnsplitss 5811	. . 3 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (dom 𝐹 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}) ⊆ 𝐹)
8	6, 7	sylanb 284	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (dom 𝐹 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}) ⊆ 𝐹)
9	5, 8	eqsstrd 3240	1 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩}) ⊆ 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1375   ∈ wcel 2180  Vcvv 2779   ∖ cdif 3174   ∪ cun 3175   ⊆ wss 3177  {csn 3646  ⟨cop 3649  dom cdm 4696   ↾ cres 4698  Rel wrel 4701  Fun wfun 5288   Fn wfn 5289  ‘cfv 5294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-br 4063  df-opab 4125  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302

This theorem is referenced by: (None)