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Theorem iunmapsn 42716
Description: The indexed union of set exponentiations to a singleton is equal to the set exponentiation of the indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
iunmapsn.x 𝑥𝜑
iunmapsn.a (𝜑𝐴𝑉)
iunmapsn.b ((𝜑𝑥𝐴) → 𝐵𝑊)
iunmapsn.c (𝜑𝐶𝑍)
Assertion
Ref Expression
iunmapsn (𝜑 𝑥𝐴 (𝐵m {𝐶}) = ( 𝑥𝐴 𝐵m {𝐶}))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑍(𝑥)

Proof of Theorem iunmapsn
Dummy variables 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iunmapsn.x . . 3 𝑥𝜑
2 iunmapsn.a . . 3 (𝜑𝐴𝑉)
3 iunmapsn.b . . 3 ((𝜑𝑥𝐴) → 𝐵𝑊)
41, 2, 3iunmapss 42714 . 2 (𝜑 𝑥𝐴 (𝐵m {𝐶}) ⊆ ( 𝑥𝐴 𝐵m {𝐶}))
5 simpr 485 . . . . . 6 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵m {𝐶})) → 𝑓 ∈ ( 𝑥𝐴 𝐵m {𝐶}))
63ex 413 . . . . . . . . . 10 (𝜑 → (𝑥𝐴𝐵𝑊))
71, 6ralrimi 3140 . . . . . . . . 9 (𝜑 → ∀𝑥𝐴 𝐵𝑊)
8 iunexg 7796 . . . . . . . . 9 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵𝑊) → 𝑥𝐴 𝐵 ∈ V)
92, 7, 8syl2anc 584 . . . . . . . 8 (𝜑 𝑥𝐴 𝐵 ∈ V)
10 iunmapsn.c . . . . . . . 8 (𝜑𝐶𝑍)
119, 10mapsnd 8662 . . . . . . 7 (𝜑 → ( 𝑥𝐴 𝐵m {𝐶}) = {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
1211adantr 481 . . . . . 6 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵m {𝐶})) → ( 𝑥𝐴 𝐵m {𝐶}) = {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
135, 12eleqtrd 2841 . . . . 5 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵m {𝐶})) → 𝑓 ∈ {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
14 abid 2719 . . . . 5 (𝑓 ∈ {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}} ↔ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
1513, 14sylib 217 . . . 4 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵m {𝐶})) → ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
16 eliun 4929 . . . . . . . . . 10 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
1716biimpi 215 . . . . . . . . 9 (𝑦 𝑥𝐴 𝐵 → ∃𝑥𝐴 𝑦𝐵)
18173ad2ant2 1133 . . . . . . . 8 ((𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑥𝐴 𝑦𝐵)
19 nfcv 2907 . . . . . . . . . . 11 𝑥𝑦
20 nfiu1 4959 . . . . . . . . . . 11 𝑥 𝑥𝐴 𝐵
2119, 20nfel 2921 . . . . . . . . . 10 𝑥 𝑦 𝑥𝐴 𝐵
22 nfv 1917 . . . . . . . . . 10 𝑥 𝑓 = {⟨𝐶, 𝑦⟩}
231, 21, 22nf3an 1904 . . . . . . . . 9 𝑥(𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
24 rspe 3235 . . . . . . . . . . . . . . . 16 ((𝑦𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
2524ancoms 459 . . . . . . . . . . . . . . 15 ((𝑓 = {⟨𝐶, 𝑦⟩} ∧ 𝑦𝐵) → ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
26 abid 2719 . . . . . . . . . . . . . . 15 (𝑓 ∈ {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} ↔ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
2725, 26sylibr 233 . . . . . . . . . . . . . 14 ((𝑓 = {⟨𝐶, 𝑦⟩} ∧ 𝑦𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
2827adantll 711 . . . . . . . . . . . . 13 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑦𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
29283adant2 1130 . . . . . . . . . . . 12 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥𝐴𝑦𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
3010adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → 𝐶𝑍)
313, 30mapsnd 8662 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴) → (𝐵m {𝐶}) = {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
3231eqcomd 2744 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵m {𝐶}))
33323adant3 1131 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴𝑦𝐵) → {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵m {𝐶}))
34333adant1r 1176 . . . . . . . . . . . 12 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥𝐴𝑦𝐵) → {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵m {𝐶}))
3529, 34eleqtrd 2841 . . . . . . . . . . 11 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥𝐴𝑦𝐵) → 𝑓 ∈ (𝐵m {𝐶}))
36353exp 1118 . . . . . . . . . 10 ((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) → (𝑥𝐴 → (𝑦𝐵𝑓 ∈ (𝐵m {𝐶}))))
37363adant2 1130 . . . . . . . . 9 ((𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → (𝑥𝐴 → (𝑦𝐵𝑓 ∈ (𝐵m {𝐶}))))
3823, 37reximdai 3242 . . . . . . . 8 ((𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → (∃𝑥𝐴 𝑦𝐵 → ∃𝑥𝐴 𝑓 ∈ (𝐵m {𝐶})))
3918, 38mpd 15 . . . . . . 7 ((𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑥𝐴 𝑓 ∈ (𝐵m {𝐶}))
40393exp 1118 . . . . . 6 (𝜑 → (𝑦 𝑥𝐴 𝐵 → (𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥𝐴 𝑓 ∈ (𝐵m {𝐶}))))
4140rexlimdv 3210 . . . . 5 (𝜑 → (∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥𝐴 𝑓 ∈ (𝐵m {𝐶})))
4241adantr 481 . . . 4 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵m {𝐶})) → (∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥𝐴 𝑓 ∈ (𝐵m {𝐶})))
4315, 42mpd 15 . . 3 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵m {𝐶})) → ∃𝑥𝐴 𝑓 ∈ (𝐵m {𝐶}))
44 eliun 4929 . . 3 (𝑓 𝑥𝐴 (𝐵m {𝐶}) ↔ ∃𝑥𝐴 𝑓 ∈ (𝐵m {𝐶}))
4543, 44sylibr 233 . 2 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵m {𝐶})) → 𝑓 𝑥𝐴 (𝐵m {𝐶}))
464, 45eqelssd 3942 1 (𝜑 𝑥𝐴 (𝐵m {𝐶}) = ( 𝑥𝐴 𝐵m {𝐶}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wnf 1786  wcel 2106  {cab 2715  wral 3064  wrex 3065  Vcvv 3430  {csn 4562  cop 4568   ciun 4925  (class class class)co 7268  m cmap 8603
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5222  ax-nul 5229  ax-pow 5287  ax-pr 5351  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3432  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-iun 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5485  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-res 5597  df-ima 5598  df-iota 6385  df-fun 6429  df-fn 6430  df-f 6431  df-f1 6432  df-fo 6433  df-f1o 6434  df-fv 6435  df-ov 7271  df-oprab 7272  df-mpo 7273  df-1st 7821  df-2nd 7822  df-map 8605
This theorem is referenced by:  ovnovollem1  44153  ovnovollem2  44154
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