Theorem resunimafz0 11096

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	⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
resunimafz0.n	⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))

Assertion

Ref	Expression
resunimafz0	⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩}))

Proof of Theorem resunimafz0

Step	Hyp	Ref	Expression
1		imaundi 5149	. . . . 5 ⊢ (𝐹 “ ((0..^𝑁) ∪ {𝑁})) = ((𝐹 “ (0..^𝑁)) ∪ (𝐹 “ {𝑁}))
2		resunimafz0.n	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))
3		elfzonn0 10426	. . . . . . . . 9 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → 𝑁 ∈ ℕ0)
4	2, 3	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
5		elnn0uz 9794	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0))
6	4, 5	sylib 122	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
7		fzisfzounsn 10483	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘0) → (0...𝑁) = ((0..^𝑁) ∪ {𝑁}))
8	6, 7	syl 14	. . . . . 6 ⊢ (𝜑 → (0...𝑁) = ((0..^𝑁) ∪ {𝑁}))
9	8	imaeq2d 5076	. . . . 5 ⊢ (𝜑 → (𝐹 “ (0...𝑁)) = (𝐹 “ ((0..^𝑁) ∪ {𝑁})))
10		resunimafz0.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
11	10	ffnd 5483	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn (0..^(♯‘𝐹)))
12		fnsnfv 5705	. . . . . . 7 ⊢ ((𝐹 Fn (0..^(♯‘𝐹)) ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → {(𝐹‘𝑁)} = (𝐹 “ {𝑁}))
13	11, 2, 12	syl2anc 411	. . . . . 6 ⊢ (𝜑 → {(𝐹‘𝑁)} = (𝐹 “ {𝑁}))
14	13	uneq2d 3361	. . . . 5 ⊢ (𝜑 → ((𝐹 “ (0..^𝑁)) ∪ {(𝐹‘𝑁)}) = ((𝐹 “ (0..^𝑁)) ∪ (𝐹 “ {𝑁})))
15	1, 9, 14	3eqtr4a 2290	. . . 4 ⊢ (𝜑 → (𝐹 “ (0...𝑁)) = ((𝐹 “ (0..^𝑁)) ∪ {(𝐹‘𝑁)}))
16	15	reseq2d 5013	. . 3 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = (𝐼 ↾ ((𝐹 “ (0..^𝑁)) ∪ {(𝐹‘𝑁)})))
17		resundi 5026	. . 3 ⊢ (𝐼 ↾ ((𝐹 “ (0..^𝑁)) ∪ {(𝐹‘𝑁)})) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ (𝐼 ↾ {(𝐹‘𝑁)}))
18	16, 17	eqtrdi 2280	. 2 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ (𝐼 ↾ {(𝐹‘𝑁)})))
19		resunimafz0.i	. . . . 5 ⊢ (𝜑 → Fun 𝐼)
20		funfn 5356	. . . . 5 ⊢ (Fun 𝐼 ↔ 𝐼 Fn dom 𝐼)
21	19, 20	sylib 122	. . . 4 ⊢ (𝜑 → 𝐼 Fn dom 𝐼)
22	10, 2	ffvelcdmd 5783	. . . 4 ⊢ (𝜑 → (𝐹‘𝑁) ∈ dom 𝐼)
23		fnressn 5840	. . . 4 ⊢ ((𝐼 Fn dom 𝐼 ∧ (𝐹‘𝑁) ∈ dom 𝐼) → (𝐼 ↾ {(𝐹‘𝑁)}) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})
24	21, 22, 23	syl2anc 411	. . 3 ⊢ (𝜑 → (𝐼 ↾ {(𝐹‘𝑁)}) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})
25	24	uneq2d 3361	. 2 ⊢ (𝜑 → ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ (𝐼 ↾ {(𝐹‘𝑁)})) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩}))
26	18, 25	eqtrd 2264	1 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩}))

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2202   ∪ cun 3198  {csn 3669  ⟨cop 3672  dom cdm 4725   ↾ cres 4727   “ cima 4728  Fun wfun 5320   Fn wfn 5321  ⟶wf 5322  ‘cfv 5326  (class class class)co 6018  0cc0 8032  ℕ₀cn0 9402  ℤ_≥cuz 9755  ...cfz 10243  ..^cfzo 10377  ♯chash 11038

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244  df-fzo 10378

This theorem is referenced by:  trlsegvdegfi  16324