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Theorem f1iun 7670
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 3402 . . . . . . . . . 10 𝑢 ∈ V
2 eqeq1 2742 . . . . . . . . . . 11 (𝑧 = 𝑢 → (𝑧 = 𝐵𝑢 = 𝐵))
32rexbidv 3207 . . . . . . . . . 10 (𝑧 = 𝑢 → (∃𝑥𝐴 𝑧 = 𝐵 ↔ ∃𝑥𝐴 𝑢 = 𝐵))
41, 3elab 3573 . . . . . . . . 9 (𝑢 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} ↔ ∃𝑥𝐴 𝑢 = 𝐵)
5 r19.29 3167 . . . . . . . . . 10 ((∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ ∃𝑥𝐴 𝑢 = 𝐵) → ∃𝑥𝐴 ((𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵))
6 nfv 1921 . . . . . . . . . . . 12 𝑥(Fun 𝑢 ∧ Fun 𝑢)
7 nfre1 3216 . . . . . . . . . . . . . 14 𝑥𝑥𝐴 𝑧 = 𝐵
87nfab 2905 . . . . . . . . . . . . 13 𝑥{𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}
9 nfv 1921 . . . . . . . . . . . . 13 𝑥(𝑢𝑣𝑣𝑢)
108, 9nfralw 3138 . . . . . . . . . . . 12 𝑥𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)
116, 10nfan 1906 . . . . . . . . . . 11 𝑥((Fun 𝑢 ∧ Fun 𝑢) ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢))
12 f1eq1 6569 . . . . . . . . . . . . . . . 16 (𝑢 = 𝐵 → (𝑢:𝐷1-1𝑆𝐵:𝐷1-1𝑆))
1312biimparc 483 . . . . . . . . . . . . . . 15 ((𝐵:𝐷1-1𝑆𝑢 = 𝐵) → 𝑢:𝐷1-1𝑆)
14 df-f1 6344 . . . . . . . . . . . . . . . 16 (𝑢:𝐷1-1𝑆 ↔ (𝑢:𝐷𝑆 ∧ Fun 𝑢))
15 ffun 6507 . . . . . . . . . . . . . . . . 17 (𝑢:𝐷𝑆 → Fun 𝑢)
1615anim1i 618 . . . . . . . . . . . . . . . 16 ((𝑢:𝐷𝑆 ∧ Fun 𝑢) → (Fun 𝑢 ∧ Fun 𝑢))
1714, 16sylbi 220 . . . . . . . . . . . . . . 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 6574 . . . . . . . . . . . . . 14 (𝐵:𝐷1-1𝑆𝐵:𝐷𝑆)
21 fiun.1 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦𝐵 = 𝐶)
2221fiunlem 7668 . . . . . . . . . . . . . 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 515 . . . . . . . . . . . 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 3225 . . . . . . . . . 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 597 . . . . . . . 8 ((∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}) → ((Fun 𝑢 ∧ Fun 𝑢) ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)))
2928ralrimiva 3096 . . . . . . 7 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ∀𝑢 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} ((Fun 𝑢 ∧ Fun 𝑢) ∧ ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢)))
30 fun11uni 7663 . . . . . . 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 498 . . . . 5 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → Fun {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵})
33 fiun.2 . . . . . . 7 𝐵 ∈ V
3433dfiun2 4919 . . . . . 6 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}
3534funeqi 6360 . . . . 5 (Fun 𝑥𝐴 𝐵 ↔ Fun {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵})
3632, 35sylibr 237 . . . 4 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → Fun 𝑥𝐴 𝐵)
371eldm2 5744 . . . . . . . . 9 (𝑢 ∈ dom 𝐵 ↔ ∃𝑣𝑢, 𝑣⟩ ∈ 𝐵)
38 f1dm 6578 . . . . . . . . . 10 (𝐵:𝐷1-1𝑆 → dom 𝐵 = 𝐷)
3938eleq2d 2818 . . . . . . . . 9 (𝐵:𝐷1-1𝑆 → (𝑢 ∈ dom 𝐵𝑢𝐷))
4037, 39bitr3id 288 . . . . . . . 8 (𝐵:𝐷1-1𝑆 → (∃𝑣𝑢, 𝑣⟩ ∈ 𝐵𝑢𝐷))
4140adantr 484 . . . . . . 7 ((𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → (∃𝑣𝑢, 𝑣⟩ ∈ 𝐵𝑢𝐷))
4241ralrexbid 3232 . . . . . 6 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → (∃𝑥𝐴𝑣𝑢, 𝑣⟩ ∈ 𝐵 ↔ ∃𝑥𝐴 𝑢𝐷))
43 eliun 4885 . . . . . . . 8 (⟨𝑢, 𝑣⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑢, 𝑣⟩ ∈ 𝐵)
4443exbii 1854 . . . . . . 7 (∃𝑣𝑢, 𝑣⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑣𝑥𝐴𝑢, 𝑣⟩ ∈ 𝐵)
451eldm2 5744 . . . . . . 7 (𝑢 ∈ dom 𝑥𝐴 𝐵 ↔ ∃𝑣𝑢, 𝑣⟩ ∈ 𝑥𝐴 𝐵)
46 rexcom4 3163 . . . . . . 7 (∃𝑥𝐴𝑣𝑢, 𝑣⟩ ∈ 𝐵 ↔ ∃𝑣𝑥𝐴𝑢, 𝑣⟩ ∈ 𝐵)
4744, 45, 463bitr4i 306 . . . . . 6 (𝑢 ∈ dom 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑣𝑢, 𝑣⟩ ∈ 𝐵)
48 eliun 4885 . . . . . 6 (𝑢 𝑥𝐴 𝐷 ↔ ∃𝑥𝐴 𝑢𝐷)
4942, 47, 483bitr4g 317 . . . . 5 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → (𝑢 ∈ dom 𝑥𝐴 𝐵𝑢 𝑥𝐴 𝐷))
5049eqrdv 2736 . . . 4 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → dom 𝑥𝐴 𝐵 = 𝑥𝐴 𝐷)
51 df-fn 6342 . . . 4 ( 𝑥𝐴 𝐵 Fn 𝑥𝐴 𝐷 ↔ (Fun 𝑥𝐴 𝐵 ∧ dom 𝑥𝐴 𝐵 = 𝑥𝐴 𝐷))
5236, 50, 51sylanbrc 586 . . 3 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 𝐵 Fn 𝑥𝐴 𝐷)
53 rniun 5980 . . . 4 ran 𝑥𝐴 𝐵 = 𝑥𝐴 ran 𝐵
5420frnd 6512 . . . . . . 7 (𝐵:𝐷1-1𝑆 → ran 𝐵𝑆)
5554adantr 484 . . . . . 6 ((𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ran 𝐵𝑆)
5655ralimi 3075 . . . . 5 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ∀𝑥𝐴 ran 𝐵𝑆)
57 iunss 4931 . . . . 5 ( 𝑥𝐴 ran 𝐵𝑆 ↔ ∀𝑥𝐴 ran 𝐵𝑆)
5856, 57sylibr 237 . . . 4 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 ran 𝐵𝑆)
5953, 58eqsstrid 3925 . . 3 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → ran 𝑥𝐴 𝐵𝑆)
60 df-f 6343 . . 3 ( 𝑥𝐴 𝐵: 𝑥𝐴 𝐷𝑆 ↔ ( 𝑥𝐴 𝐵 Fn 𝑥𝐴 𝐷 ∧ ran 𝑥𝐴 𝐵𝑆))
6152, 59, 60sylanbrc 586 . 2 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 𝐵: 𝑥𝐴 𝐷𝑆)
6231simprd 499 . . 3 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → Fun {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵})
6334cnveqi 5717 . . . 4 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}
6463funeqi 6360 . . 3 (Fun 𝑥𝐴 𝐵 ↔ Fun {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵})
6562, 64sylibr 237 . 2 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → Fun 𝑥𝐴 𝐵)
66 df-f1 6344 . 2 ( 𝑥𝐴 𝐵: 𝑥𝐴 𝐷1-1𝑆 ↔ ( 𝑥𝐴 𝐵: 𝑥𝐴 𝐷𝑆 ∧ Fun 𝑥𝐴 𝐵))
6761, 65, 66sylanbrc 586 1 (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 𝐵: 𝑥𝐴 𝐷1-1𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 846   = wceq 1542  wex 1786  wcel 2114  {cab 2716  wral 3053  wrex 3054  Vcvv 3398  wss 3843  cop 4522   cuni 4796   ciun 4881  ccnv 5524  dom cdm 5525  ran crn 5526  Fun wfun 6333   Fn wfn 6334  wf 6335  1-1wf1 6336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344
This theorem is referenced by:  ackbij2  9743
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