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Theorem iunmapsn 39900
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 39898 . 2 (𝜑 𝑥𝐴 (𝐵𝑚 {𝐶}) ⊆ ( 𝑥𝐴 𝐵𝑚 {𝐶}))
5 simpr 479 . . . . . . . 8 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})) → 𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶}))
63ex 449 . . . . . . . . . . . 12 (𝜑 → (𝑥𝐴𝐵𝑊))
71, 6ralrimi 3087 . . . . . . . . . . 11 (𝜑 → ∀𝑥𝐴 𝐵𝑊)
8 iunexg 7300 . . . . . . . . . . 11 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵𝑊) → 𝑥𝐴 𝐵 ∈ V)
92, 7, 8syl2anc 696 . . . . . . . . . 10 (𝜑 𝑥𝐴 𝐵 ∈ V)
10 iunmapsn.c . . . . . . . . . 10 (𝜑𝐶𝑍)
119, 10mapsnd 39879 . . . . . . . . 9 (𝜑 → ( 𝑥𝐴 𝐵𝑚 {𝐶}) = {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
1211adantr 472 . . . . . . . 8 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})) → ( 𝑥𝐴 𝐵𝑚 {𝐶}) = {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
135, 12eleqtrd 2833 . . . . . . 7 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})) → 𝑓 ∈ {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
14 abid 2740 . . . . . . 7 (𝑓 ∈ {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}} ↔ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
1513, 14sylib 208 . . . . . 6 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})) → ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
16 eliun 4668 . . . . . . . . . . . 12 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
1716biimpi 206 . . . . . . . . . . 11 (𝑦 𝑥𝐴 𝐵 → ∃𝑥𝐴 𝑦𝐵)
18173ad2ant2 1128 . . . . . . . . . 10 ((𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑥𝐴 𝑦𝐵)
19 nfcv 2894 . . . . . . . . . . . . 13 𝑥𝑦
20 nfiu1 4694 . . . . . . . . . . . . 13 𝑥 𝑥𝐴 𝐵
2119, 20nfel 2907 . . . . . . . . . . . 12 𝑥 𝑦 𝑥𝐴 𝐵
22 nfv 1984 . . . . . . . . . . . 12 𝑥 𝑓 = {⟨𝐶, 𝑦⟩}
231, 21, 22nf3an 1972 . . . . . . . . . . 11 𝑥(𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
24 rspe 3133 . . . . . . . . . . . . . . . . . 18 ((𝑦𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
2524ancoms 468 . . . . . . . . . . . . . . . . 17 ((𝑓 = {⟨𝐶, 𝑦⟩} ∧ 𝑦𝐵) → ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
26 abid 2740 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} ↔ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
2725, 26sylibr 224 . . . . . . . . . . . . . . . 16 ((𝑓 = {⟨𝐶, 𝑦⟩} ∧ 𝑦𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
2827adantll 752 . . . . . . . . . . . . . . 15 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑦𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
29283adant2 1125 . . . . . . . . . . . . . 14 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥𝐴𝑦𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
3010adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → 𝐶𝑍)
313, 30mapsnd 39879 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (𝐵𝑚 {𝐶}) = {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
3231eqcomd 2758 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵𝑚 {𝐶}))
33323adant3 1126 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑦𝐵) → {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵𝑚 {𝐶}))
34333adant1r 1185 . . . . . . . . . . . . . 14 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥𝐴𝑦𝐵) → {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵𝑚 {𝐶}))
3529, 34eleqtrd 2833 . . . . . . . . . . . . 13 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥𝐴𝑦𝐵) → 𝑓 ∈ (𝐵𝑚 {𝐶}))
36353exp 1112 . . . . . . . . . . . 12 ((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) → (𝑥𝐴 → (𝑦𝐵𝑓 ∈ (𝐵𝑚 {𝐶}))))
37363adant2 1125 . . . . . . . . . . 11 ((𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → (𝑥𝐴 → (𝑦𝐵𝑓 ∈ (𝐵𝑚 {𝐶}))))
3823, 37reximdai 3142 . . . . . . . . . 10 ((𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → (∃𝑥𝐴 𝑦𝐵 → ∃𝑥𝐴 𝑓 ∈ (𝐵𝑚 {𝐶})))
3918, 38mpd 15 . . . . . . . . 9 ((𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑥𝐴 𝑓 ∈ (𝐵𝑚 {𝐶}))
40393exp 1112 . . . . . . . 8 (𝜑 → (𝑦 𝑥𝐴 𝐵 → (𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥𝐴 𝑓 ∈ (𝐵𝑚 {𝐶}))))
4140rexlimdv 3160 . . . . . . 7 (𝜑 → (∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥𝐴 𝑓 ∈ (𝐵𝑚 {𝐶})))
4241adantr 472 . . . . . 6 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})) → (∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥𝐴 𝑓 ∈ (𝐵𝑚 {𝐶})))
4315, 42mpd 15 . . . . 5 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})) → ∃𝑥𝐴 𝑓 ∈ (𝐵𝑚 {𝐶}))
44 eliun 4668 . . . . 5 (𝑓 𝑥𝐴 (𝐵𝑚 {𝐶}) ↔ ∃𝑥𝐴 𝑓 ∈ (𝐵𝑚 {𝐶}))
4543, 44sylibr 224 . . . 4 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})) → 𝑓 𝑥𝐴 (𝐵𝑚 {𝐶}))
4645ralrimiva 3096 . . 3 (𝜑 → ∀𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})𝑓 𝑥𝐴 (𝐵𝑚 {𝐶}))
47 dfss3 3725 . . 3 (( 𝑥𝐴 𝐵𝑚 {𝐶}) ⊆ 𝑥𝐴 (𝐵𝑚 {𝐶}) ↔ ∀𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})𝑓 𝑥𝐴 (𝐵𝑚 {𝐶}))
4846, 47sylibr 224 . 2 (𝜑 → ( 𝑥𝐴 𝐵𝑚 {𝐶}) ⊆ 𝑥𝐴 (𝐵𝑚 {𝐶}))
494, 48eqssd 3753 1 (𝜑 𝑥𝐴 (𝐵𝑚 {𝐶}) = ( 𝑥𝐴 𝐵𝑚 {𝐶}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1624  wnf 1849  wcel 2131  {cab 2738  wral 3042  wrex 3043  Vcvv 3332  wss 3707  {csn 4313  cop 4319   ciun 4664  (class class class)co 6805  𝑚 cmap 8015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-1st 7325  df-2nd 7326  df-map 8017
This theorem is referenced by:  ovnovollem1  41368  ovnovollem2  41369
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