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Theorem fiun 7967
Description: The union of a chain (with respect to inclusion) of functions is a function. Analogous to f1iun 7968. (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 3484 . . . . . . . 8 𝑢 ∈ V
2 eqeq1 2741 . . . . . . . . 9 (𝑧 = 𝑢 → (𝑧 = 𝐵𝑢 = 𝐵))
32rexbidv 3179 . . . . . . . 8 (𝑧 = 𝑢 → (∃𝑥𝐴 𝑧 = 𝐵 ↔ ∃𝑥𝐴 𝑢 = 𝐵))
41, 3elab 3679 . . . . . . 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 1914 . . . . . . . . . 10 𝑥Fun 𝑢
7 nfre1 3285 . . . . . . . . . . . 12 𝑥𝑥𝐴 𝑧 = 𝐵
87nfab 2911 . . . . . . . . . . 11 𝑥{𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}
9 nfv 1914 . . . . . . . . . . 11 𝑥(𝑢𝑣𝑣𝑢)
108, 9nfralw 3311 . . . . . . . . . 10 𝑥𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)
116, 10nfan 1899 . . . . . . . . 9 𝑥(Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢))
12 ffun 6739 . . . . . . . . . . . . 13 (𝐵:𝐷𝑆 → Fun 𝐵)
13 funeq 6586 . . . . . . . . . . . . 13 (𝑢 = 𝐵 → (Fun 𝑢 ↔ Fun 𝐵))
14 bianir 1059 . . . . . . . . . . . . 13 ((Fun 𝐵 ∧ (Fun 𝑢 ↔ Fun 𝐵)) → Fun 𝑢)
1512, 13, 14syl2an 596 . . . . . . . . . . . 12 ((𝐵:𝐷𝑆𝑢 = 𝐵) → Fun 𝑢)
1615adantlr 715 . . . . . . . . . . 11 (((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) → Fun 𝑢)
17 fiun.1 . . . . . . . . . . . 12 (𝑥 = 𝑦𝐵 = 𝐶)
1817fiunlem 7966 . . . . . . . . . . 11 (((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) → ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢))
1916, 18jca 511 . . . . . . . . . 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 3259 . . . . . . . 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 594 . . . . . 6 ((∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}) → (Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)))
2423ralrimiva 3146 . . . . 5 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ∀𝑢 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)))
25 fununi 6641 . . . . 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 5033 . . . . 5 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}
2928funeqi 6587 . . . 4 (Fun 𝑥𝐴 𝐵 ↔ Fun {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵})
3026, 29sylibr 234 . . 3 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → Fun 𝑥𝐴 𝐵)
311eldm2 5912 . . . . . . . 8 (𝑢 ∈ dom 𝐵 ↔ ∃𝑣𝑢, 𝑣⟩ ∈ 𝐵)
32 fdm 6745 . . . . . . . . 9 (𝐵:𝐷𝑆 → dom 𝐵 = 𝐷)
3332eleq2d 2827 . . . . . . . 8 (𝐵:𝐷𝑆 → (𝑢 ∈ dom 𝐵𝑢𝐷))
3431, 33bitr3id 285 . . . . . . 7 (𝐵:𝐷𝑆 → (∃𝑣𝑢, 𝑣⟩ ∈ 𝐵𝑢𝐷))
3534adantr 480 . . . . . 6 ((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → (∃𝑣𝑢, 𝑣⟩ ∈ 𝐵𝑢𝐷))
3635ralrexbid 3106 . . . . 5 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → (∃𝑥𝐴𝑣𝑢, 𝑣⟩ ∈ 𝐵 ↔ ∃𝑥𝐴 𝑢𝐷))
37 eliun 4995 . . . . . . 7 (⟨𝑢, 𝑣⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑢, 𝑣⟩ ∈ 𝐵)
3837exbii 1848 . . . . . 6 (∃𝑣𝑢, 𝑣⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑣𝑥𝐴𝑢, 𝑣⟩ ∈ 𝐵)
391eldm2 5912 . . . . . 6 (𝑢 ∈ dom 𝑥𝐴 𝐵 ↔ ∃𝑣𝑢, 𝑣⟩ ∈ 𝑥𝐴 𝐵)
40 rexcom4 3288 . . . . . 6 (∃𝑥𝐴𝑣𝑢, 𝑣⟩ ∈ 𝐵 ↔ ∃𝑣𝑥𝐴𝑢, 𝑣⟩ ∈ 𝐵)
4138, 39, 403bitr4i 303 . . . . 5 (𝑢 ∈ dom 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑣𝑢, 𝑣⟩ ∈ 𝐵)
42 eliun 4995 . . . . 5 (𝑢 𝑥𝐴 𝐷 ↔ ∃𝑥𝐴 𝑢𝐷)
4336, 41, 423bitr4g 314 . . . 4 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → (𝑢 ∈ dom 𝑥𝐴 𝐵𝑢 𝑥𝐴 𝐷))
4443eqrdv 2735 . . 3 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → dom 𝑥𝐴 𝐵 = 𝑥𝐴 𝐷)
45 df-fn 6564 . . 3 ( 𝑥𝐴 𝐵 Fn 𝑥𝐴 𝐷 ↔ (Fun 𝑥𝐴 𝐵 ∧ dom 𝑥𝐴 𝐵 = 𝑥𝐴 𝐷))
4630, 44, 45sylanbrc 583 . 2 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 𝐵 Fn 𝑥𝐴 𝐷)
47 rniun 6167 . . 3 ran 𝑥𝐴 𝐵 = 𝑥𝐴 ran 𝐵
48 frn 6743 . . . . . 6 (𝐵:𝐷𝑆 → ran 𝐵𝑆)
4948adantr 480 . . . . 5 ((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ran 𝐵𝑆)
5049ralimi 3083 . . . 4 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ∀𝑥𝐴 ran 𝐵𝑆)
51 iunss 5045 . . . 4 ( 𝑥𝐴 ran 𝐵𝑆 ↔ ∀𝑥𝐴 ran 𝐵𝑆)
5250, 51sylibr 234 . . 3 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 ran 𝐵𝑆)
5347, 52eqsstrid 4022 . 2 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ran 𝑥𝐴 𝐵𝑆)
54 df-f 6565 . 2 ( 𝑥𝐴 𝐵: 𝑥𝐴 𝐷𝑆 ↔ ( 𝑥𝐴 𝐵 Fn 𝑥𝐴 𝐷 ∧ ran 𝑥𝐴 𝐵𝑆))
5546, 53, 54sylanbrc 583 1 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 𝐵: 𝑥𝐴 𝐷𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wral 3061  wrex 3070  Vcvv 3480  wss 3951  cop 4632   cuni 4907   ciun 4991  dom cdm 5685  ran crn 5686  Fun wfun 6555   Fn wfn 6556  wf 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-f 6565
This theorem is referenced by:  satfun  35416
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