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Theorem iunmapsn 43916
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 43914 . 2 (𝜑 𝑥𝐴 (𝐵m {𝐶}) ⊆ ( 𝑥𝐴 𝐵m {𝐶}))
5 simpr 486 . . . . . 6 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵m {𝐶})) → 𝑓 ∈ ( 𝑥𝐴 𝐵m {𝐶}))
63ex 414 . . . . . . . . . 10 (𝜑 → (𝑥𝐴𝐵𝑊))
71, 6ralrimi 3255 . . . . . . . . 9 (𝜑 → ∀𝑥𝐴 𝐵𝑊)
8 iunexg 7950 . . . . . . . . 9 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵𝑊) → 𝑥𝐴 𝐵 ∈ V)
92, 7, 8syl2anc 585 . . . . . . . 8 (𝜑 𝑥𝐴 𝐵 ∈ V)
10 iunmapsn.c . . . . . . . 8 (𝜑𝐶𝑍)
119, 10mapsnd 8880 . . . . . . 7 (𝜑 → ( 𝑥𝐴 𝐵m {𝐶}) = {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
1211adantr 482 . . . . . 6 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵m {𝐶})) → ( 𝑥𝐴 𝐵m {𝐶}) = {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
135, 12eleqtrd 2836 . . . . 5 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵m {𝐶})) → 𝑓 ∈ {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
14 abid 2714 . . . . 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 5002 . . . . . . . . . 10 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
1716biimpi 215 . . . . . . . . 9 (𝑦 𝑥𝐴 𝐵 → ∃𝑥𝐴 𝑦𝐵)
18173ad2ant2 1135 . . . . . . . 8 ((𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑥𝐴 𝑦𝐵)
19 nfcv 2904 . . . . . . . . . . 11 𝑥𝑦
20 nfiu1 5032 . . . . . . . . . . 11 𝑥 𝑥𝐴 𝐵
2119, 20nfel 2918 . . . . . . . . . 10 𝑥 𝑦 𝑥𝐴 𝐵
22 nfv 1918 . . . . . . . . . 10 𝑥 𝑓 = {⟨𝐶, 𝑦⟩}
231, 21, 22nf3an 1905 . . . . . . . . 9 𝑥(𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
24 rspe 3247 . . . . . . . . . . . . . . . 16 ((𝑦𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
2524ancoms 460 . . . . . . . . . . . . . . 15 ((𝑓 = {⟨𝐶, 𝑦⟩} ∧ 𝑦𝐵) → ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
26 abid 2714 . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . 13 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑦𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
29283adant2 1132 . . . . . . . . . . . 12 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥𝐴𝑦𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
3010adantr 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → 𝐶𝑍)
313, 30mapsnd 8880 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴) → (𝐵m {𝐶}) = {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
3231eqcomd 2739 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵m {𝐶}))
33323adant3 1133 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴𝑦𝐵) → {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵m {𝐶}))
34333adant1r 1178 . . . . . . . . . . . 12 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥𝐴𝑦𝐵) → {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵m {𝐶}))
3529, 34eleqtrd 2836 . . . . . . . . . . 11 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥𝐴𝑦𝐵) → 𝑓 ∈ (𝐵m {𝐶}))
36353exp 1120 . . . . . . . . . 10 ((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) → (𝑥𝐴 → (𝑦𝐵𝑓 ∈ (𝐵m {𝐶}))))
37363adant2 1132 . . . . . . . . 9 ((𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → (𝑥𝐴 → (𝑦𝐵𝑓 ∈ (𝐵m {𝐶}))))
3823, 37reximdai 3259 . . . . . . . 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 1120 . . . . . 6 (𝜑 → (𝑦 𝑥𝐴 𝐵 → (𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥𝐴 𝑓 ∈ (𝐵m {𝐶}))))
4140rexlimdv 3154 . . . . 5 (𝜑 → (∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥𝐴 𝑓 ∈ (𝐵m {𝐶})))
4241adantr 482 . . . 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 5002 . . 3 (𝑓 𝑥𝐴 (𝐵m {𝐶}) ↔ ∃𝑥𝐴 𝑓 ∈ (𝐵m {𝐶}))
4543, 44sylibr 233 . 2 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵m {𝐶})) → 𝑓 𝑥𝐴 (𝐵m {𝐶}))
464, 45eqelssd 4004 1 (𝜑 𝑥𝐴 (𝐵m {𝐶}) = ( 𝑥𝐴 𝐵m {𝐶}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wnf 1786  wcel 2107  {cab 2710  wral 3062  wrex 3071  Vcvv 3475  {csn 4629  cop 4635   ciun 4998  (class class class)co 7409  m cmap 8820
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822
This theorem is referenced by:  ovnovollem1  45372  ovnovollem2  45373
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