MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fiun Structured version   Visualization version   GIF version

Theorem fiun 7876
Description: The union of a chain (with respect to inclusion) of functions is a function. Analogous to f1iun 7877. (Contributed by AV, 6-Oct-2023.)
Hypotheses
Ref Expression
fiun.1 (𝑥 = 𝑦𝐵 = 𝐶)
fiun.2 𝐵 ∈ V
Assertion
Ref Expression
fiun (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 𝐵: 𝑥𝐴 𝐷𝑆)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐶   𝑥,𝑦   𝑥,𝑆
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)   𝐷(𝑥,𝑦)   𝑆(𝑦)

Proof of Theorem fiun
Dummy variables 𝑣 𝑧 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3448 . . . . . . . 8 𝑢 ∈ V
2 eqeq1 2737 . . . . . . . . 9 (𝑧 = 𝑢 → (𝑧 = 𝐵𝑢 = 𝐵))
32rexbidv 3172 . . . . . . . 8 (𝑧 = 𝑢 → (∃𝑥𝐴 𝑧 = 𝐵 ↔ ∃𝑥𝐴 𝑢 = 𝐵))
41, 3elab 3631 . . . . . . 7 (𝑢 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} ↔ ∃𝑥𝐴 𝑢 = 𝐵)
5 r19.29 3114 . . . . . . . 8 ((∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ ∃𝑥𝐴 𝑢 = 𝐵) → ∃𝑥𝐴 ((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵))
6 nfv 1918 . . . . . . . . . 10 𝑥Fun 𝑢
7 nfre1 3267 . . . . . . . . . . . 12 𝑥𝑥𝐴 𝑧 = 𝐵
87nfab 2910 . . . . . . . . . . 11 𝑥{𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}
9 nfv 1918 . . . . . . . . . . 11 𝑥(𝑢𝑣𝑣𝑢)
108, 9nfralw 3293 . . . . . . . . . 10 𝑥𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)
116, 10nfan 1903 . . . . . . . . 9 𝑥(Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢))
12 ffun 6672 . . . . . . . . . . . . 13 (𝐵:𝐷𝑆 → Fun 𝐵)
13 funeq 6522 . . . . . . . . . . . . 13 (𝑢 = 𝐵 → (Fun 𝑢 ↔ Fun 𝐵))
14 bianir 1058 . . . . . . . . . . . . 13 ((Fun 𝐵 ∧ (Fun 𝑢 ↔ Fun 𝐵)) → Fun 𝑢)
1512, 13, 14syl2an 597 . . . . . . . . . . . 12 ((𝐵:𝐷𝑆𝑢 = 𝐵) → Fun 𝑢)
1615adantlr 714 . . . . . . . . . . 11 (((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) → Fun 𝑢)
17 fiun.1 . . . . . . . . . . . 12 (𝑥 = 𝑦𝐵 = 𝐶)
1817fiunlem 7875 . . . . . . . . . . 11 (((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) → ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢))
1916, 18jca 513 . . . . . . . . . 10 (((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) → (Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)))
2019a1i 11 . . . . . . . . 9 (𝑥𝐴 → (((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) → (Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢))))
2111, 20rexlimi 3241 . . . . . . . 8 (∃𝑥𝐴 ((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) → (Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)))
225, 21syl 17 . . . . . . 7 ((∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ ∃𝑥𝐴 𝑢 = 𝐵) → (Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)))
234, 22sylan2b 595 . . . . . 6 ((∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}) → (Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)))
2423ralrimiva 3140 . . . . 5 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ∀𝑢 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)))
25 fununi 6577 . . . . 5 (∀𝑢 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)) → Fun {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵})
2624, 25syl 17 . . . 4 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → Fun {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵})
27 fiun.2 . . . . . 6 𝐵 ∈ V
2827dfiun2 4994 . . . . 5 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}
2928funeqi 6523 . . . 4 (Fun 𝑥𝐴 𝐵 ↔ Fun {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵})
3026, 29sylibr 233 . . 3 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → Fun 𝑥𝐴 𝐵)
311eldm2 5858 . . . . . . . 8 (𝑢 ∈ dom 𝐵 ↔ ∃𝑣𝑢, 𝑣⟩ ∈ 𝐵)
32 fdm 6678 . . . . . . . . 9 (𝐵:𝐷𝑆 → dom 𝐵 = 𝐷)
3332eleq2d 2820 . . . . . . . 8 (𝐵:𝐷𝑆 → (𝑢 ∈ dom 𝐵𝑢𝐷))
3431, 33bitr3id 285 . . . . . . 7 (𝐵:𝐷𝑆 → (∃𝑣𝑢, 𝑣⟩ ∈ 𝐵𝑢𝐷))
3534adantr 482 . . . . . 6 ((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → (∃𝑣𝑢, 𝑣⟩ ∈ 𝐵𝑢𝐷))
3635ralrexbid 3106 . . . . 5 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → (∃𝑥𝐴𝑣𝑢, 𝑣⟩ ∈ 𝐵 ↔ ∃𝑥𝐴 𝑢𝐷))
37 eliun 4959 . . . . . . 7 (⟨𝑢, 𝑣⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑢, 𝑣⟩ ∈ 𝐵)
3837exbii 1851 . . . . . 6 (∃𝑣𝑢, 𝑣⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑣𝑥𝐴𝑢, 𝑣⟩ ∈ 𝐵)
391eldm2 5858 . . . . . 6 (𝑢 ∈ dom 𝑥𝐴 𝐵 ↔ ∃𝑣𝑢, 𝑣⟩ ∈ 𝑥𝐴 𝐵)
40 rexcom4 3270 . . . . . 6 (∃𝑥𝐴𝑣𝑢, 𝑣⟩ ∈ 𝐵 ↔ ∃𝑣𝑥𝐴𝑢, 𝑣⟩ ∈ 𝐵)
4138, 39, 403bitr4i 303 . . . . 5 (𝑢 ∈ dom 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑣𝑢, 𝑣⟩ ∈ 𝐵)
42 eliun 4959 . . . . 5 (𝑢 𝑥𝐴 𝐷 ↔ ∃𝑥𝐴 𝑢𝐷)
4336, 41, 423bitr4g 314 . . . 4 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → (𝑢 ∈ dom 𝑥𝐴 𝐵𝑢 𝑥𝐴 𝐷))
4443eqrdv 2731 . . 3 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → dom 𝑥𝐴 𝐵 = 𝑥𝐴 𝐷)
45 df-fn 6500 . . 3 ( 𝑥𝐴 𝐵 Fn 𝑥𝐴 𝐷 ↔ (Fun 𝑥𝐴 𝐵 ∧ dom 𝑥𝐴 𝐵 = 𝑥𝐴 𝐷))
4630, 44, 45sylanbrc 584 . 2 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 𝐵 Fn 𝑥𝐴 𝐷)
47 rniun 6101 . . 3 ran 𝑥𝐴 𝐵 = 𝑥𝐴 ran 𝐵
48 frn 6676 . . . . . 6 (𝐵:𝐷𝑆 → ran 𝐵𝑆)
4948adantr 482 . . . . 5 ((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ran 𝐵𝑆)
5049ralimi 3083 . . . 4 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ∀𝑥𝐴 ran 𝐵𝑆)
51 iunss 5006 . . . 4 ( 𝑥𝐴 ran 𝐵𝑆 ↔ ∀𝑥𝐴 ran 𝐵𝑆)
5250, 51sylibr 233 . . 3 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 ran 𝐵𝑆)
5347, 52eqsstrid 3993 . 2 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ran 𝑥𝐴 𝐵𝑆)
54 df-f 6501 . 2 ( 𝑥𝐴 𝐵: 𝑥𝐴 𝐷𝑆 ↔ ( 𝑥𝐴 𝐵 Fn 𝑥𝐴 𝐷 ∧ ran 𝑥𝐴 𝐵𝑆))
5546, 53, 54sylanbrc 584 1 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 𝐵: 𝑥𝐴 𝐷𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wral 3061  wrex 3070  Vcvv 3444  wss 3911  cop 4593   cuni 4866   ciun 4955  dom cdm 5634  ran crn 5635  Fun wfun 6491   Fn wfn 6492  wf 6493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-fun 6499  df-fn 6500  df-f 6501
This theorem is referenced by:  satfun  34062
  Copyright terms: Public domain W3C validator