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Theorem iunmapsn 38404
 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 38402 . 2 (𝜑 𝑥𝐴 (𝐵𝑚 {𝐶}) ⊆ ( 𝑥𝐴 𝐵𝑚 {𝐶}))
5 simpr 476 . . . . . . . 8 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})) → 𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶}))
63ex 449 . . . . . . . . . . . 12 (𝜑 → (𝑥𝐴𝐵𝑊))
71, 6ralrimi 2940 . . . . . . . . . . 11 (𝜑 → ∀𝑥𝐴 𝐵𝑊)
8 iunexg 7035 . . . . . . . . . . 11 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵𝑊) → 𝑥𝐴 𝐵 ∈ V)
92, 7, 8syl2anc 691 . . . . . . . . . 10 (𝜑 𝑥𝐴 𝐵 ∈ V)
10 iunmapsn.c . . . . . . . . . 10 (𝜑𝐶𝑍)
119, 10mapsnd 38383 . . . . . . . . 9 (𝜑 → ( 𝑥𝐴 𝐵𝑚 {𝐶}) = {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
1211adantr 480 . . . . . . . 8 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})) → ( 𝑥𝐴 𝐵𝑚 {𝐶}) = {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
135, 12eleqtrd 2690 . . . . . . 7 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})) → 𝑓 ∈ {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
14 abid 2598 . . . . . . 7 (𝑓 ∈ {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}} ↔ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
1513, 14sylib 207 . . . . . 6 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})) → ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
16 eliun 4460 . . . . . . . . . . . 12 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
1716biimpi 205 . . . . . . . . . . 11 (𝑦 𝑥𝐴 𝐵 → ∃𝑥𝐴 𝑦𝐵)
18173ad2ant2 1076 . . . . . . . . . 10 ((𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑥𝐴 𝑦𝐵)
19 nfcv 2751 . . . . . . . . . . . . 13 𝑥𝑦
20 nfiu1 4486 . . . . . . . . . . . . 13 𝑥 𝑥𝐴 𝐵
2119, 20nfel 2763 . . . . . . . . . . . 12 𝑥 𝑦 𝑥𝐴 𝐵
22 nfv 1830 . . . . . . . . . . . 12 𝑥 𝑓 = {⟨𝐶, 𝑦⟩}
231, 21, 22nf3an 1819 . . . . . . . . . . 11 𝑥(𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
24 rspe 2986 . . . . . . . . . . . . . . . . . 18 ((𝑦𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
2524ancoms 468 . . . . . . . . . . . . . . . . 17 ((𝑓 = {⟨𝐶, 𝑦⟩} ∧ 𝑦𝐵) → ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
26 abid 2598 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} ↔ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
2725, 26sylibr 223 . . . . . . . . . . . . . . . 16 ((𝑓 = {⟨𝐶, 𝑦⟩} ∧ 𝑦𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
2827adantll 746 . . . . . . . . . . . . . . 15 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑦𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
29283adant2 1073 . . . . . . . . . . . . . 14 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥𝐴𝑦𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
3010adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → 𝐶𝑍)
313, 30mapsnd 38383 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (𝐵𝑚 {𝐶}) = {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
3231eqcomd 2616 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵𝑚 {𝐶}))
33323adant3 1074 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴𝑦𝐵) → {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵𝑚 {𝐶}))
34333adant1r 1311 . . . . . . . . . . . . . 14 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥𝐴𝑦𝐵) → {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵𝑚 {𝐶}))
3529, 34eleqtrd 2690 . . . . . . . . . . . . 13 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥𝐴𝑦𝐵) → 𝑓 ∈ (𝐵𝑚 {𝐶}))
36353exp 1256 . . . . . . . . . . . 12 ((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) → (𝑥𝐴 → (𝑦𝐵𝑓 ∈ (𝐵𝑚 {𝐶}))))
37363adant2 1073 . . . . . . . . . . 11 ((𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → (𝑥𝐴 → (𝑦𝐵𝑓 ∈ (𝐵𝑚 {𝐶}))))
3823, 37reximdai 2995 . . . . . . . . . 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 1256 . . . . . . . 8 (𝜑 → (𝑦 𝑥𝐴 𝐵 → (𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥𝐴 𝑓 ∈ (𝐵𝑚 {𝐶}))))
4140rexlimdv 3012 . . . . . . 7 (𝜑 → (∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥𝐴 𝑓 ∈ (𝐵𝑚 {𝐶})))
4241adantr 480 . . . . . 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 4460 . . . . 5 (𝑓 𝑥𝐴 (𝐵𝑚 {𝐶}) ↔ ∃𝑥𝐴 𝑓 ∈ (𝐵𝑚 {𝐶}))
4543, 44sylibr 223 . . . 4 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})) → 𝑓 𝑥𝐴 (𝐵𝑚 {𝐶}))
4645ralrimiva 2949 . . 3 (𝜑 → ∀𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})𝑓 𝑥𝐴 (𝐵𝑚 {𝐶}))
47 dfss3 3558 . . 3 (( 𝑥𝐴 𝐵𝑚 {𝐶}) ⊆ 𝑥𝐴 (𝐵𝑚 {𝐶}) ↔ ∀𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})𝑓 𝑥𝐴 (𝐵𝑚 {𝐶}))
4846, 47sylibr 223 . 2 (𝜑 → ( 𝑥𝐴 𝐵𝑚 {𝐶}) ⊆ 𝑥𝐴 (𝐵𝑚 {𝐶}))
494, 48eqssd 3585 1 (𝜑 𝑥𝐴 (𝐵𝑚 {𝐶}) = ( 𝑥𝐴 𝐵𝑚 {𝐶}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  {csn 4125  ⟨cop 4131  ∪ ciun 4455  (class class class)co 6549   ↑𝑚 cmap 7744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746 This theorem is referenced by:  ovnovollem1  39546  ovnovollem2  39547
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