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Theorem resunimafz0 10795
Description: The union of a restriction by an image over an open range of nonnegative integers and a singleton of an ordered pair is a restriction by an image over an interval of nonnegative integers. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.)
Hypotheses
Ref Expression
resunimafz0.i (𝜑 → Fun 𝐼)
resunimafz0.f (𝜑𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
resunimafz0.n (𝜑𝑁 ∈ (0..^(♯‘𝐹)))
Assertion
Ref Expression
resunimafz0 (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))

Proof of Theorem resunimafz0
StepHypRef Expression
1 imaundi 5037 . . . . 5 (𝐹 “ ((0..^𝑁) ∪ {𝑁})) = ((𝐹 “ (0..^𝑁)) ∪ (𝐹 “ {𝑁}))
2 resunimafz0.n . . . . . . . . 9 (𝜑𝑁 ∈ (0..^(♯‘𝐹)))
3 elfzonn0 10172 . . . . . . . . 9 (𝑁 ∈ (0..^(♯‘𝐹)) → 𝑁 ∈ ℕ0)
42, 3syl 14 . . . . . . . 8 (𝜑𝑁 ∈ ℕ0)
5 elnn0uz 9554 . . . . . . . 8 (𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))
64, 5sylib 122 . . . . . . 7 (𝜑𝑁 ∈ (ℤ‘0))
7 fzisfzounsn 10222 . . . . . . 7 (𝑁 ∈ (ℤ‘0) → (0...𝑁) = ((0..^𝑁) ∪ {𝑁}))
86, 7syl 14 . . . . . 6 (𝜑 → (0...𝑁) = ((0..^𝑁) ∪ {𝑁}))
98imaeq2d 4966 . . . . 5 (𝜑 → (𝐹 “ (0...𝑁)) = (𝐹 “ ((0..^𝑁) ∪ {𝑁})))
10 resunimafz0.f . . . . . . . 8 (𝜑𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
1110ffnd 5362 . . . . . . 7 (𝜑𝐹 Fn (0..^(♯‘𝐹)))
12 fnsnfv 5571 . . . . . . 7 ((𝐹 Fn (0..^(♯‘𝐹)) ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → {(𝐹𝑁)} = (𝐹 “ {𝑁}))
1311, 2, 12syl2anc 411 . . . . . 6 (𝜑 → {(𝐹𝑁)} = (𝐹 “ {𝑁}))
1413uneq2d 3289 . . . . 5 (𝜑 → ((𝐹 “ (0..^𝑁)) ∪ {(𝐹𝑁)}) = ((𝐹 “ (0..^𝑁)) ∪ (𝐹 “ {𝑁})))
151, 9, 143eqtr4a 2236 . . . 4 (𝜑 → (𝐹 “ (0...𝑁)) = ((𝐹 “ (0..^𝑁)) ∪ {(𝐹𝑁)}))
1615reseq2d 4903 . . 3 (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = (𝐼 ↾ ((𝐹 “ (0..^𝑁)) ∪ {(𝐹𝑁)})))
17 resundi 4916 . . 3 (𝐼 ↾ ((𝐹 “ (0..^𝑁)) ∪ {(𝐹𝑁)})) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ (𝐼 ↾ {(𝐹𝑁)}))
1816, 17eqtrdi 2226 . 2 (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ (𝐼 ↾ {(𝐹𝑁)})))
19 resunimafz0.i . . . . 5 (𝜑 → Fun 𝐼)
20 funfn 5242 . . . . 5 (Fun 𝐼𝐼 Fn dom 𝐼)
2119, 20sylib 122 . . . 4 (𝜑𝐼 Fn dom 𝐼)
2210, 2ffvelcdmd 5648 . . . 4 (𝜑 → (𝐹𝑁) ∈ dom 𝐼)
23 fnressn 5698 . . . 4 ((𝐼 Fn dom 𝐼 ∧ (𝐹𝑁) ∈ dom 𝐼) → (𝐼 ↾ {(𝐹𝑁)}) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})
2421, 22, 23syl2anc 411 . . 3 (𝜑 → (𝐼 ↾ {(𝐹𝑁)}) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})
2524uneq2d 3289 . 2 (𝜑 → ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ (𝐼 ↾ {(𝐹𝑁)})) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))
2618, 25eqtrd 2210 1 (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  cun 3127  {csn 3591  cop 3594  dom cdm 4623  cres 4625  cima 4626  Fun wfun 5206   Fn wfn 5207  wf 5208  cfv 5212  (class class class)co 5869  0cc0 7802  0cn0 9165  cuz 9517  ...cfz 9995  ..^cfzo 10128  chash 10739
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518  df-fz 9996  df-fzo 10129
This theorem is referenced by: (None)
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