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Theorem iunmapsn 38901
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 38899 . 2 (𝜑 𝑥𝐴 (𝐵𝑚 {𝐶}) ⊆ ( 𝑥𝐴 𝐵𝑚 {𝐶}))
5 simpr 477 . . . . . . . 8 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})) → 𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶}))
63ex 450 . . . . . . . . . . . 12 (𝜑 → (𝑥𝐴𝐵𝑊))
71, 6ralrimi 2951 . . . . . . . . . . 11 (𝜑 → ∀𝑥𝐴 𝐵𝑊)
8 iunexg 7092 . . . . . . . . . . 11 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵𝑊) → 𝑥𝐴 𝐵 ∈ V)
92, 7, 8syl2anc 692 . . . . . . . . . 10 (𝜑 𝑥𝐴 𝐵 ∈ V)
10 iunmapsn.c . . . . . . . . . 10 (𝜑𝐶𝑍)
119, 10mapsnd 38880 . . . . . . . . 9 (𝜑 → ( 𝑥𝐴 𝐵𝑚 {𝐶}) = {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
1211adantr 481 . . . . . . . 8 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})) → ( 𝑥𝐴 𝐵𝑚 {𝐶}) = {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
135, 12eleqtrd 2700 . . . . . . 7 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})) → 𝑓 ∈ {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
14 abid 2609 . . . . . . 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 4492 . . . . . . . . . . . 12 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
1716biimpi 206 . . . . . . . . . . 11 (𝑦 𝑥𝐴 𝐵 → ∃𝑥𝐴 𝑦𝐵)
18173ad2ant2 1081 . . . . . . . . . 10 ((𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑥𝐴 𝑦𝐵)
19 nfcv 2761 . . . . . . . . . . . . 13 𝑥𝑦
20 nfiu1 4518 . . . . . . . . . . . . 13 𝑥 𝑥𝐴 𝐵
2119, 20nfel 2773 . . . . . . . . . . . 12 𝑥 𝑦 𝑥𝐴 𝐵
22 nfv 1840 . . . . . . . . . . . 12 𝑥 𝑓 = {⟨𝐶, 𝑦⟩}
231, 21, 22nf3an 1828 . . . . . . . . . . 11 𝑥(𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
24 rspe 2997 . . . . . . . . . . . . . . . . . 18 ((𝑦𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
2524ancoms 469 . . . . . . . . . . . . . . . . 17 ((𝑓 = {⟨𝐶, 𝑦⟩} ∧ 𝑦𝐵) → ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
26 abid 2609 . . . . . . . . . . . . . . . . 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 749 . . . . . . . . . . . . . . 15 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑦𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
29283adant2 1078 . . . . . . . . . . . . . 14 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥𝐴𝑦𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
3010adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → 𝐶𝑍)
313, 30mapsnd 38880 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (𝐵𝑚 {𝐶}) = {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
3231eqcomd 2627 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵𝑚 {𝐶}))
33323adant3 1079 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑦𝐵) → {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵𝑚 {𝐶}))
34333adant1r 1316 . . . . . . . . . . . . . 14 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥𝐴𝑦𝐵) → {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵𝑚 {𝐶}))
3529, 34eleqtrd 2700 . . . . . . . . . . . . 13 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥𝐴𝑦𝐵) → 𝑓 ∈ (𝐵𝑚 {𝐶}))
36353exp 1261 . . . . . . . . . . . 12 ((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) → (𝑥𝐴 → (𝑦𝐵𝑓 ∈ (𝐵𝑚 {𝐶}))))
37363adant2 1078 . . . . . . . . . . 11 ((𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → (𝑥𝐴 → (𝑦𝐵𝑓 ∈ (𝐵𝑚 {𝐶}))))
3823, 37reximdai 3006 . . . . . . . . . 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 1261 . . . . . . . 8 (𝜑 → (𝑦 𝑥𝐴 𝐵 → (𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥𝐴 𝑓 ∈ (𝐵𝑚 {𝐶}))))
4140rexlimdv 3023 . . . . . . 7 (𝜑 → (∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥𝐴 𝑓 ∈ (𝐵𝑚 {𝐶})))
4241adantr 481 . . . . . 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 4492 . . . . 5 (𝑓 𝑥𝐴 (𝐵𝑚 {𝐶}) ↔ ∃𝑥𝐴 𝑓 ∈ (𝐵𝑚 {𝐶}))
4543, 44sylibr 224 . . . 4 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})) → 𝑓 𝑥𝐴 (𝐵𝑚 {𝐶}))
4645ralrimiva 2960 . . 3 (𝜑 → ∀𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})𝑓 𝑥𝐴 (𝐵𝑚 {𝐶}))
47 dfss3 3574 . . 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 3601 1 (𝜑 𝑥𝐴 (𝐵𝑚 {𝐶}) = ( 𝑥𝐴 𝐵𝑚 {𝐶}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wnf 1705  wcel 1987  {cab 2607  wral 2907  wrex 2908  Vcvv 3186  wss 3556  {csn 4150  cop 4156   ciun 4487  (class class class)co 6607  𝑚 cmap 7805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-1st 7116  df-2nd 7117  df-map 7807
This theorem is referenced by:  ovnovollem1  40193  ovnovollem2  40194
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