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Theorem fnsnsplitss 5535
Description: Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 20-Jan-2023.)
Assertion
Ref Expression
fnsnsplitss ((𝐹 Fn 𝐴𝑋𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}) ⊆ 𝐹)

Proof of Theorem fnsnsplitss
StepHypRef Expression
1 difsnss 3605 . . . 4 (𝑋𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ⊆ 𝐴)
21adantl 272 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ⊆ 𝐴)
3 ssres2 4772 . . 3 (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ⊆ 𝐴 → (𝐹 ↾ ((𝐴 ∖ {𝑋}) ∪ {𝑋})) ⊆ (𝐹𝐴))
42, 3syl 14 . 2 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹 ↾ ((𝐴 ∖ {𝑋}) ∪ {𝑋})) ⊆ (𝐹𝐴))
5 resundi 4758 . . 3 (𝐹 ↾ ((𝐴 ∖ {𝑋}) ∪ {𝑋})) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ (𝐹 ↾ {𝑋}))
6 fnressn 5522 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹 ↾ {𝑋}) = {⟨𝑋, (𝐹𝑋)⟩})
76uneq2d 3169 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ (𝐹 ↾ {𝑋})) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))
85, 7syl5eq 2139 . 2 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹 ↾ ((𝐴 ∖ {𝑋}) ∪ {𝑋})) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))
9 fnresdm 5157 . . 3 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
109adantr 271 . 2 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹𝐴) = 𝐹)
114, 8, 103sstr3d 3083 1 ((𝐹 Fn 𝐴𝑋𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}) ⊆ 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1296  wcel 1445  cdif 3010  cun 3011  wss 3013  {csn 3466  cop 3469  cres 4469   Fn wfn 5044  cfv 5049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-reu 2377  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057
This theorem is referenced by:  funresdfunsnss  5539
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