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Theorem fiun 7965
Description: The union of a chain (with respect to inclusion) of functions is a function. Analogous to f1iun 7966. (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 3481 . . . . . . . 8 𝑢 ∈ V
2 eqeq1 2738 . . . . . . . . 9 (𝑧 = 𝑢 → (𝑧 = 𝐵𝑢 = 𝐵))
32rexbidv 3176 . . . . . . . 8 (𝑧 = 𝑢 → (∃𝑥𝐴 𝑧 = 𝐵 ↔ ∃𝑥𝐴 𝑢 = 𝐵))
41, 3elab 3680 . . . . . . 7 (𝑢 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} ↔ ∃𝑥𝐴 𝑢 = 𝐵)
5 r19.29 3111 . . . . . . . 8 ((∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ ∃𝑥𝐴 𝑢 = 𝐵) → ∃𝑥𝐴 ((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵))
6 nfv 1911 . . . . . . . . . 10 𝑥Fun 𝑢
7 nfre1 3282 . . . . . . . . . . . 12 𝑥𝑥𝐴 𝑧 = 𝐵
87nfab 2908 . . . . . . . . . . 11 𝑥{𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}
9 nfv 1911 . . . . . . . . . . 11 𝑥(𝑢𝑣𝑣𝑢)
108, 9nfralw 3308 . . . . . . . . . 10 𝑥𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)
116, 10nfan 1896 . . . . . . . . 9 𝑥(Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢))
12 ffun 6739 . . . . . . . . . . . . 13 (𝐵:𝐷𝑆 → Fun 𝐵)
13 funeq 6587 . . . . . . . . . . . . 13 (𝑢 = 𝐵 → (Fun 𝑢 ↔ Fun 𝐵))
14 bianir 1058 . . . . . . . . . . . . 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 7964 . . . . . . . . . . 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 3256 . . . . . . . 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 3143 . . . . 5 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ∀𝑢 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)))
25 fununi 6642 . . . . 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 5037 . . . . 5 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}
2928funeqi 6588 . . . 4 (Fun 𝑥𝐴 𝐵 ↔ Fun {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵})
3026, 29sylibr 234 . . 3 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → Fun 𝑥𝐴 𝐵)
311eldm2 5914 . . . . . . . 8 (𝑢 ∈ dom 𝐵 ↔ ∃𝑣𝑢, 𝑣⟩ ∈ 𝐵)
32 fdm 6745 . . . . . . . . 9 (𝐵:𝐷𝑆 → dom 𝐵 = 𝐷)
3332eleq2d 2824 . . . . . . . 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 3103 . . . . 5 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → (∃𝑥𝐴𝑣𝑢, 𝑣⟩ ∈ 𝐵 ↔ ∃𝑥𝐴 𝑢𝐷))
37 eliun 4999 . . . . . . 7 (⟨𝑢, 𝑣⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑢, 𝑣⟩ ∈ 𝐵)
3837exbii 1844 . . . . . 6 (∃𝑣𝑢, 𝑣⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑣𝑥𝐴𝑢, 𝑣⟩ ∈ 𝐵)
391eldm2 5914 . . . . . 6 (𝑢 ∈ dom 𝑥𝐴 𝐵 ↔ ∃𝑣𝑢, 𝑣⟩ ∈ 𝑥𝐴 𝐵)
40 rexcom4 3285 . . . . . 6 (∃𝑥𝐴𝑣𝑢, 𝑣⟩ ∈ 𝐵 ↔ ∃𝑣𝑥𝐴𝑢, 𝑣⟩ ∈ 𝐵)
4138, 39, 403bitr4i 303 . . . . 5 (𝑢 ∈ dom 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑣𝑢, 𝑣⟩ ∈ 𝐵)
42 eliun 4999 . . . . 5 (𝑢 𝑥𝐴 𝐷 ↔ ∃𝑥𝐴 𝑢𝐷)
4336, 41, 423bitr4g 314 . . . 4 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → (𝑢 ∈ dom 𝑥𝐴 𝐵𝑢 𝑥𝐴 𝐷))
4443eqrdv 2732 . . 3 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → dom 𝑥𝐴 𝐵 = 𝑥𝐴 𝐷)
45 df-fn 6565 . . 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 6169 . . 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 3080 . . . 4 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ∀𝑥𝐴 ran 𝐵𝑆)
51 iunss 5049 . . . 4 ( 𝑥𝐴 ran 𝐵𝑆 ↔ ∀𝑥𝐴 ran 𝐵𝑆)
5250, 51sylibr 234 . . 3 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 ran 𝐵𝑆)
5347, 52eqsstrid 4043 . 2 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ran 𝑥𝐴 𝐵𝑆)
54 df-f 6566 . 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 847   = wceq 1536  wex 1775  wcel 2105  {cab 2711  wral 3058  wrex 3067  Vcvv 3477  wss 3962  cop 4636   cuni 4911   ciun 4995  dom cdm 5688  ran crn 5689  Fun wfun 6556   Fn wfn 6557  wf 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-fun 6564  df-fn 6565  df-f 6566
This theorem is referenced by:  satfun  35395
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