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Theorem f1iun 7966
Description: The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by Mario Carneiro, 20-May-2013.) (Revised by Mario Carneiro, 24-Jun-2015.) (Proof shortened by AV, 5-Nov-2023.)
Hypotheses
Ref Expression
fiun.1 (𝑥 = 𝑦𝐵 = 𝐶)
fiun.2 𝐵 ∈ V
Assertion
Ref Expression
f1iun (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 𝐵: 𝑥𝐴 𝐷1-1𝑆)
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 f1iun
Dummy variables 𝑣 𝑧 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3481 . . . . . . . . . 10 𝑢 ∈ V
2 eqeq1 2738 . . . . . . . . . . 11 (𝑧 = 𝑢 → (𝑧 = 𝐵𝑢 = 𝐵))
32rexbidv 3176 . . . . . . . . . 10 (𝑧 = 𝑢 → (∃𝑥𝐴 𝑧 = 𝐵 ↔ ∃𝑥𝐴 𝑢 = 𝐵))
41, 3elab 3680 . . . . . . . . 9 (𝑢 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} ↔ ∃𝑥𝐴 𝑢 = 𝐵)
5 r19.29 3111 . . . . . . . . . 10 ((∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ ∃𝑥𝐴 𝑢 = 𝐵) → ∃𝑥𝐴 ((𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵))
6 nfv 1911 . . . . . . . . . . . 12 𝑥(Fun 𝑢 ∧ Fun 𝑢)
7 nfre1 3282 . . . . . . . . . . . . . 14 𝑥𝑥𝐴 𝑧 = 𝐵
87nfab 2908 . . . . . . . . . . . . 13 𝑥{𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}
9 nfv 1911 . . . . . . . . . . . . 13 𝑥(𝑢𝑣𝑣𝑢)
108, 9nfralw 3308 . . . . . . . . . . . 12 𝑥𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)
116, 10nfan 1896 . . . . . . . . . . 11 𝑥((Fun 𝑢 ∧ Fun 𝑢) ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢))
12 f1eq1 6799 . . . . . . . . . . . . . . . 16 (𝑢 = 𝐵 → (𝑢:𝐷1-1𝑆𝐵:𝐷1-1𝑆))
1312biimparc 479 . . . . . . . . . . . . . . 15 ((𝐵:𝐷1-1𝑆𝑢 = 𝐵) → 𝑢:𝐷1-1𝑆)
14 df-f1 6567 . . . . . . . . . . . . . . . 16 (𝑢:𝐷1-1𝑆 ↔ (𝑢:𝐷𝑆 ∧ Fun 𝑢))
15 ffun 6739 . . . . . . . . . . . . . . . . 17 (𝑢:𝐷𝑆 → Fun 𝑢)
1615anim1i 615 . . . . . . . . . . . . . . . 16 ((𝑢:𝐷𝑆 ∧ Fun 𝑢) → (Fun 𝑢 ∧ Fun 𝑢))
1714, 16sylbi 217 . . . . . . . . . . . . . . 15 (𝑢:𝐷1-1𝑆 → (Fun 𝑢 ∧ Fun 𝑢))
1813, 17syl 17 . . . . . . . . . . . . . 14 ((𝐵:𝐷1-1𝑆𝑢 = 𝐵) → (Fun 𝑢 ∧ Fun 𝑢))
1918adantlr 715 . . . . . . . . . . . . 13 (((𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) → (Fun 𝑢 ∧ Fun 𝑢))
20 f1f 6804 . . . . . . . . . . . . . 14 (𝐵:𝐷1-1𝑆𝐵:𝐷𝑆)
21 fiun.1 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦𝐵 = 𝐶)
2221fiunlem 7964 . . . . . . . . . . . . . 14 (((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) → ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢))
2320, 22sylanl1 680 . . . . . . . . . . . . 13 (((𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) → ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢))
2419, 23jca 511 . . . . . . . . . . . 12 (((𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) → ((Fun 𝑢 ∧ Fun 𝑢) ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)))
2524a1i 11 . . . . . . . . . . 11 (𝑥𝐴 → (((𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) → ((Fun 𝑢 ∧ Fun 𝑢) ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢))))
2611, 25rexlimi 3256 . . . . . . . . . 10 (∃𝑥𝐴 ((𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) → ((Fun 𝑢 ∧ Fun 𝑢) ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)))
275, 26syl 17 . . . . . . . . 9 ((∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ ∃𝑥𝐴 𝑢 = 𝐵) → ((Fun 𝑢 ∧ Fun 𝑢) ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)))
284, 27sylan2b 594 . . . . . . . 8 ((∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}) → ((Fun 𝑢 ∧ Fun 𝑢) ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)))
2928ralrimiva 3143 . . . . . . 7 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ∀𝑢 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} ((Fun 𝑢 ∧ Fun 𝑢) ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)))
30 fun11uni 7955 . . . . . . 7 (∀𝑢 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} ((Fun 𝑢 ∧ Fun 𝑢) ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)) → (Fun {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} ∧ Fun {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}))
3129, 30syl 17 . . . . . 6 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → (Fun {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} ∧ Fun {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}))
3231simpld 494 . . . . 5 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → Fun {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵})
33 fiun.2 . . . . . . 7 𝐵 ∈ V
3433dfiun2 5037 . . . . . 6 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}
3534funeqi 6588 . . . . 5 (Fun 𝑥𝐴 𝐵 ↔ Fun {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵})
3632, 35sylibr 234 . . . 4 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → Fun 𝑥𝐴 𝐵)
371eldm2 5914 . . . . . . . . 9 (𝑢 ∈ dom 𝐵 ↔ ∃𝑣𝑢, 𝑣⟩ ∈ 𝐵)
38 f1dm 6808 . . . . . . . . . 10 (𝐵:𝐷1-1𝑆 → dom 𝐵 = 𝐷)
3938eleq2d 2824 . . . . . . . . 9 (𝐵:𝐷1-1𝑆 → (𝑢 ∈ dom 𝐵𝑢𝐷))
4037, 39bitr3id 285 . . . . . . . 8 (𝐵:𝐷1-1𝑆 → (∃𝑣𝑢, 𝑣⟩ ∈ 𝐵𝑢𝐷))
4140adantr 480 . . . . . . 7 ((𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → (∃𝑣𝑢, 𝑣⟩ ∈ 𝐵𝑢𝐷))
4241ralrexbid 3103 . . . . . 6 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → (∃𝑥𝐴𝑣𝑢, 𝑣⟩ ∈ 𝐵 ↔ ∃𝑥𝐴 𝑢𝐷))
43 eliun 4999 . . . . . . . 8 (⟨𝑢, 𝑣⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑢, 𝑣⟩ ∈ 𝐵)
4443exbii 1844 . . . . . . 7 (∃𝑣𝑢, 𝑣⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑣𝑥𝐴𝑢, 𝑣⟩ ∈ 𝐵)
451eldm2 5914 . . . . . . 7 (𝑢 ∈ dom 𝑥𝐴 𝐵 ↔ ∃𝑣𝑢, 𝑣⟩ ∈ 𝑥𝐴 𝐵)
46 rexcom4 3285 . . . . . . 7 (∃𝑥𝐴𝑣𝑢, 𝑣⟩ ∈ 𝐵 ↔ ∃𝑣𝑥𝐴𝑢, 𝑣⟩ ∈ 𝐵)
4744, 45, 463bitr4i 303 . . . . . 6 (𝑢 ∈ dom 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑣𝑢, 𝑣⟩ ∈ 𝐵)
48 eliun 4999 . . . . . 6 (𝑢 𝑥𝐴 𝐷 ↔ ∃𝑥𝐴 𝑢𝐷)
4942, 47, 483bitr4g 314 . . . . 5 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → (𝑢 ∈ dom 𝑥𝐴 𝐵𝑢 𝑥𝐴 𝐷))
5049eqrdv 2732 . . . 4 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → dom 𝑥𝐴 𝐵 = 𝑥𝐴 𝐷)
51 df-fn 6565 . . . 4 ( 𝑥𝐴 𝐵 Fn 𝑥𝐴 𝐷 ↔ (Fun 𝑥𝐴 𝐵 ∧ dom 𝑥𝐴 𝐵 = 𝑥𝐴 𝐷))
5236, 50, 51sylanbrc 583 . . 3 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 𝐵 Fn 𝑥𝐴 𝐷)
53 rniun 6169 . . . 4 ran 𝑥𝐴 𝐵 = 𝑥𝐴 ran 𝐵
5420frnd 6744 . . . . . . 7 (𝐵:𝐷1-1𝑆 → ran 𝐵𝑆)
5554adantr 480 . . . . . 6 ((𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ran 𝐵𝑆)
5655ralimi 3080 . . . . 5 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ∀𝑥𝐴 ran 𝐵𝑆)
57 iunss 5049 . . . . 5 ( 𝑥𝐴 ran 𝐵𝑆 ↔ ∀𝑥𝐴 ran 𝐵𝑆)
5856, 57sylibr 234 . . . 4 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 ran 𝐵𝑆)
5953, 58eqsstrid 4043 . . 3 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ran 𝑥𝐴 𝐵𝑆)
60 df-f 6566 . . 3 ( 𝑥𝐴 𝐵: 𝑥𝐴 𝐷𝑆 ↔ ( 𝑥𝐴 𝐵 Fn 𝑥𝐴 𝐷 ∧ ran 𝑥𝐴 𝐵𝑆))
6152, 59, 60sylanbrc 583 . 2 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 𝐵: 𝑥𝐴 𝐷𝑆)
6231simprd 495 . . 3 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → Fun {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵})
6334cnveqi 5887 . . . 4 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}
6463funeqi 6588 . . 3 (Fun 𝑥𝐴 𝐵 ↔ Fun {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵})
6562, 64sylibr 234 . 2 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → Fun 𝑥𝐴 𝐵)
66 df-f1 6567 . 2 ( 𝑥𝐴 𝐵: 𝑥𝐴 𝐷1-1𝑆 ↔ ( 𝑥𝐴 𝐵: 𝑥𝐴 𝐷𝑆 ∧ Fun 𝑥𝐴 𝐵))
6761, 65, 66sylanbrc 583 1 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 𝐵: 𝑥𝐴 𝐷1-1𝑆)
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  ccnv 5687  dom cdm 5688  ran crn 5689  Fun wfun 6556   Fn wfn 6557  wf 6558  1-1wf1 6559
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-pow 5370  ax-pr 5437  ax-un 7753
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-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  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  df-f1 6567
This theorem is referenced by:  ackbij2  10279
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