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Theorem iunmapsn 39896
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 39894 . 2 (𝜑 𝑥𝐴 (𝐵𝑚 {𝐶}) ⊆ ( 𝑥𝐴 𝐵𝑚 {𝐶}))
5 simpr 473 . . . . . 6 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})) → 𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶}))
63ex 399 . . . . . . . . . 10 (𝜑 → (𝑥𝐴𝐵𝑊))
71, 6ralrimi 3145 . . . . . . . . 9 (𝜑 → ∀𝑥𝐴 𝐵𝑊)
8 iunexg 7373 . . . . . . . . 9 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵𝑊) → 𝑥𝐴 𝐵 ∈ V)
92, 7, 8syl2anc 575 . . . . . . . 8 (𝜑 𝑥𝐴 𝐵 ∈ V)
10 iunmapsn.c . . . . . . . 8 (𝜑𝐶𝑍)
119, 10mapsnd 8134 . . . . . . 7 (𝜑 → ( 𝑥𝐴 𝐵𝑚 {𝐶}) = {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
1211adantr 468 . . . . . 6 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})) → ( 𝑥𝐴 𝐵𝑚 {𝐶}) = {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
135, 12eleqtrd 2887 . . . . 5 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})) → 𝑓 ∈ {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}})
14 abid 2794 . . . . 5 (𝑓 ∈ {𝑓 ∣ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}} ↔ ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
1513, 14sylib 209 . . . 4 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})) → ∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
16 eliun 4716 . . . . . . . . . 10 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
1716biimpi 207 . . . . . . . . 9 (𝑦 𝑥𝐴 𝐵 → ∃𝑥𝐴 𝑦𝐵)
18173ad2ant2 1157 . . . . . . . 8 ((𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑥𝐴 𝑦𝐵)
19 nfcv 2948 . . . . . . . . . . 11 𝑥𝑦
20 nfiu1 4742 . . . . . . . . . . 11 𝑥 𝑥𝐴 𝐵
2119, 20nfel 2961 . . . . . . . . . 10 𝑥 𝑦 𝑥𝐴 𝐵
22 nfv 2005 . . . . . . . . . 10 𝑥 𝑓 = {⟨𝐶, 𝑦⟩}
231, 21, 22nf3an 1993 . . . . . . . . 9 𝑥(𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩})
24 rspe 3190 . . . . . . . . . . . . . . . 16 ((𝑦𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
2524ancoms 448 . . . . . . . . . . . . . . 15 ((𝑓 = {⟨𝐶, 𝑦⟩} ∧ 𝑦𝐵) → ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
26 abid 2794 . . . . . . . . . . . . . . 15 (𝑓 ∈ {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} ↔ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩})
2725, 26sylibr 225 . . . . . . . . . . . . . 14 ((𝑓 = {⟨𝐶, 𝑦⟩} ∧ 𝑦𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
2827adantll 696 . . . . . . . . . . . . 13 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑦𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
29283adant2 1154 . . . . . . . . . . . 12 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥𝐴𝑦𝐵) → 𝑓 ∈ {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
3010adantr 468 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → 𝐶𝑍)
313, 30mapsnd 8134 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴) → (𝐵𝑚 {𝐶}) = {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}})
3231eqcomd 2812 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵𝑚 {𝐶}))
33323adant3 1155 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴𝑦𝐵) → {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵𝑚 {𝐶}))
34333adant1r 1216 . . . . . . . . . . . 12 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥𝐴𝑦𝐵) → {𝑓 ∣ ∃𝑦𝐵 𝑓 = {⟨𝐶, 𝑦⟩}} = (𝐵𝑚 {𝐶}))
3529, 34eleqtrd 2887 . . . . . . . . . . 11 (((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) ∧ 𝑥𝐴𝑦𝐵) → 𝑓 ∈ (𝐵𝑚 {𝐶}))
36353exp 1141 . . . . . . . . . 10 ((𝜑𝑓 = {⟨𝐶, 𝑦⟩}) → (𝑥𝐴 → (𝑦𝐵𝑓 ∈ (𝐵𝑚 {𝐶}))))
37363adant2 1154 . . . . . . . . 9 ((𝜑𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩}) → (𝑥𝐴 → (𝑦𝐵𝑓 ∈ (𝐵𝑚 {𝐶}))))
3823, 37reximdai 3199 . . . . . . . 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 1141 . . . . . 6 (𝜑 → (𝑦 𝑥𝐴 𝐵 → (𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥𝐴 𝑓 ∈ (𝐵𝑚 {𝐶}))))
4140rexlimdv 3218 . . . . 5 (𝜑 → (∃𝑦 𝑥𝐴 𝐵𝑓 = {⟨𝐶, 𝑦⟩} → ∃𝑥𝐴 𝑓 ∈ (𝐵𝑚 {𝐶})))
4241adantr 468 . . . 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 4716 . . 3 (𝑓 𝑥𝐴 (𝐵𝑚 {𝐶}) ↔ ∃𝑥𝐴 𝑓 ∈ (𝐵𝑚 {𝐶}))
4543, 44sylibr 225 . 2 ((𝜑𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 {𝐶})) → 𝑓 𝑥𝐴 (𝐵𝑚 {𝐶}))
464, 45eqelssd 3819 1 (𝜑 𝑥𝐴 (𝐵𝑚 {𝐶}) = ( 𝑥𝐴 𝐵𝑚 {𝐶}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100   = wceq 1637  wnf 1863  wcel 2156  {cab 2792  wral 3096  wrex 3097  Vcvv 3391  {csn 4370  cop 4376   ciun 4712  (class class class)co 6874  𝑚 cmap 8092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-1st 7398  df-2nd 7399  df-map 8094
This theorem is referenced by:  ovnovollem1  41352  ovnovollem2  41353
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