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Theorem fiun 7929
Description: The union of a chain (with respect to inclusion) of functions is a function. Analogous to f1iun 7930. (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 3479 . . . . . . . 8 𝑢 ∈ V
2 eqeq1 2737 . . . . . . . . 9 (𝑧 = 𝑢 → (𝑧 = 𝐵𝑢 = 𝐵))
32rexbidv 3179 . . . . . . . 8 (𝑧 = 𝑢 → (∃𝑥𝐴 𝑧 = 𝐵 ↔ ∃𝑥𝐴 𝑢 = 𝐵))
41, 3elab 3669 . . . . . . 7 (𝑢 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} ↔ ∃𝑥𝐴 𝑢 = 𝐵)
5 r19.29 3115 . . . . . . . 8 ((∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ ∃𝑥𝐴 𝑢 = 𝐵) → ∃𝑥𝐴 ((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵))
6 nfv 1918 . . . . . . . . . 10 𝑥Fun 𝑢
7 nfre1 3283 . . . . . . . . . . . 12 𝑥𝑥𝐴 𝑧 = 𝐵
87nfab 2910 . . . . . . . . . . 11 𝑥{𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}
9 nfv 1918 . . . . . . . . . . 11 𝑥(𝑢𝑣𝑣𝑢)
108, 9nfralw 3309 . . . . . . . . . 10 𝑥𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)
116, 10nfan 1903 . . . . . . . . 9 𝑥(Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢))
12 ffun 6721 . . . . . . . . . . . . 13 (𝐵:𝐷𝑆 → Fun 𝐵)
13 funeq 6569 . . . . . . . . . . . . 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 7928 . . . . . . . . . . 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 3257 . . . . . . . 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 3147 . . . . 5 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ∀𝑢 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (Fun 𝑢 ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)))
25 fununi 6624 . . . . 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 6570 . . . 4 (Fun 𝑥𝐴 𝐵 ↔ Fun {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵})
3026, 29sylibr 233 . . 3 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → Fun 𝑥𝐴 𝐵)
311eldm2 5902 . . . . . . . 8 (𝑢 ∈ dom 𝐵 ↔ ∃𝑣𝑢, 𝑣⟩ ∈ 𝐵)
32 fdm 6727 . . . . . . . . 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 3107 . . . . 5 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → (∃𝑥𝐴𝑣𝑢, 𝑣⟩ ∈ 𝐵 ↔ ∃𝑥𝐴 𝑢𝐷))
37 eliun 5002 . . . . . . 7 (⟨𝑢, 𝑣⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑢, 𝑣⟩ ∈ 𝐵)
3837exbii 1851 . . . . . 6 (∃𝑣𝑢, 𝑣⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑣𝑥𝐴𝑢, 𝑣⟩ ∈ 𝐵)
391eldm2 5902 . . . . . 6 (𝑢 ∈ dom 𝑥𝐴 𝐵 ↔ ∃𝑣𝑢, 𝑣⟩ ∈ 𝑥𝐴 𝐵)
40 rexcom4 3286 . . . . . 6 (∃𝑥𝐴𝑣𝑢, 𝑣⟩ ∈ 𝐵 ↔ ∃𝑣𝑥𝐴𝑢, 𝑣⟩ ∈ 𝐵)
4138, 39, 403bitr4i 303 . . . . 5 (𝑢 ∈ dom 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑣𝑢, 𝑣⟩ ∈ 𝐵)
42 eliun 5002 . . . . 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 6547 . . 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 6148 . . 3 ran 𝑥𝐴 𝐵 = 𝑥𝐴 ran 𝐵
48 frn 6725 . . . . . 6 (𝐵:𝐷𝑆 → ran 𝐵𝑆)
4948adantr 482 . . . . 5 ((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ran 𝐵𝑆)
5049ralimi 3084 . . . 4 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ∀𝑥𝐴 ran 𝐵𝑆)
51 iunss 5049 . . . 4 ( 𝑥𝐴 ran 𝐵𝑆 ↔ ∀𝑥𝐴 ran 𝐵𝑆)
5250, 51sylibr 233 . . 3 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 ran 𝐵𝑆)
5347, 52eqsstrid 4031 . 2 (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ran 𝑥𝐴 𝐵𝑆)
54 df-f 6548 . 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 3062  wrex 3071  Vcvv 3475  wss 3949  cop 4635   cuni 4909   ciun 4998  dom cdm 5677  ran crn 5678  Fun wfun 6538   Fn wfn 6539  wf 6540
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 5300  ax-nul 5307  ax-pr 5428
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 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-fun 6546  df-fn 6547  df-f 6548
This theorem is referenced by:  satfun  34402
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