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Theorem iunmapsn 44761
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 44759 . 2 (𝜑 𝑥𝐴 (𝐵m {𝐶}) ⊆ ( 𝑥𝐴 𝐵m {𝐶}))
5 simpr 483 . . . . . 6 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵m {𝐶})) → 𝑓 ∈ ( 𝑥𝐴 𝐵m {𝐶}))
63ex 411 . . . . . . . . . 10 (𝜑 → (𝑥𝐴𝐵𝑊))
71, 6ralrimi 3244 . . . . . . . . 9 (𝜑 → ∀𝑥𝐴 𝐵𝑊)
8 iunexg 7976 . . . . . . . . 9 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵𝑊) → 𝑥𝐴 𝐵 ∈ V)
92, 7, 8syl2anc 582 . . . . . . . 8 (𝜑 𝑥𝐴 𝐵 ∈ V)
10 iunmapsn.c . . . . . . . 8 (𝜑𝐶𝑍)
119, 10mapsnd 8914 . . . . . . 7 (𝜑 → ( 𝑥𝐴 𝐵m {𝐶}) = {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
1211adantr 479 . . . . . 6 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵m {𝐶})) → ( 𝑥𝐴 𝐵m {𝐶}) = {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
135, 12eleqtrd 2827 . . . . 5 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵m {𝐶})) → 𝑓 ∈ {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
14 abid 2706 . . . . 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 5004 . . . . . . . . . 10 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
1716biimpi 215 . . . . . . . . 9 (𝑦 𝑥𝐴 𝐵 → ∃𝑥𝐴 𝑦𝐵)
18173ad2ant2 1131 . . . . . . . 8 ((𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑥𝐴 𝑦𝐵)
19 nfcv 2891 . . . . . . . . . . 11 𝑥𝑦
20 nfiu1 5034 . . . . . . . . . . 11 𝑥 𝑥𝐴 𝐵
2119, 20nfel 2906 . . . . . . . . . 10 𝑥 𝑦 𝑥𝐴 𝐵
22 nfv 1909 . . . . . . . . . 10 𝑥 𝑓 = {⟨𝐶, 𝑦⟩}
231, 21, 22nf3an 1896 . . . . . . . . 9 𝑥(𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
24 rspe 3236 . . . . . . . . . . . . . . . 16 ((𝑦𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
2524ancoms 457 . . . . . . . . . . . . . . 15 ((𝑓 = {⟨𝐶, 𝑦⟩} ∧ 𝑦𝐵) → ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
26 abid 2706 . . . . . . . . . . . . . . 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 712 . . . . . . . . . . . . 13 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑦𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
29283adant2 1128 . . . . . . . . . . . 12 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥𝐴𝑦𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
3010adantr 479 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → 𝐶𝑍)
313, 30mapsnd 8914 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴) → (𝐵m {𝐶}) = {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
3231eqcomd 2731 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵m {𝐶}))
33323adant3 1129 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴𝑦𝐵) → {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵m {𝐶}))
34333adant1r 1174 . . . . . . . . . . . 12 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥𝐴𝑦𝐵) → {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵m {𝐶}))
3529, 34eleqtrd 2827 . . . . . . . . . . 11 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥𝐴𝑦𝐵) → 𝑓 ∈ (𝐵m {𝐶}))
36353exp 1116 . . . . . . . . . 10 ((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) → (𝑥𝐴 → (𝑦𝐵𝑓 ∈ (𝐵m {𝐶}))))
37363adant2 1128 . . . . . . . . 9 ((𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → (𝑥𝐴 → (𝑦𝐵𝑓 ∈ (𝐵m {𝐶}))))
3823, 37reximdai 3248 . . . . . . . 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 1116 . . . . . 6 (𝜑 → (𝑦 𝑥𝐴 𝐵 → (𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥𝐴 𝑓 ∈ (𝐵m {𝐶}))))
4140rexlimdv 3142 . . . . 5 (𝜑 → (∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥𝐴 𝑓 ∈ (𝐵m {𝐶})))
4241adantr 479 . . . 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 5004 . . 3 (𝑓 𝑥𝐴 (𝐵m {𝐶}) ↔ ∃𝑥𝐴 𝑓 ∈ (𝐵m {𝐶}))
4543, 44sylibr 233 . 2 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵m {𝐶})) → 𝑓 𝑥𝐴 (𝐵m {𝐶}))
464, 45eqelssd 4000 1 (𝜑 𝑥𝐴 (𝐵m {𝐶}) = ( 𝑥𝐴 𝐵m {𝐶}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wnf 1777  wcel 2098  {cab 2702  wral 3050  wrex 3059  Vcvv 3461  {csn 4632  cop 4638   ciun 5000  (class class class)co 7423  m cmap 8854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 8002  df-2nd 8003  df-map 8856
This theorem is referenced by:  ovnovollem1  46214  ovnovollem2  46215
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