Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  unirnmapsn Structured version   Visualization version   GIF version

Theorem unirnmapsn 42276
Description: Equality theorem for a subset of a set exponentiation, where the exponent is a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
unirnmapsn.A (𝜑𝐴𝑉)
unirnmapsn.b (𝜑𝐵𝑊)
unirnmapsn.C 𝐶 = {𝐴}
unirnmapsn.x (𝜑𝑋 ⊆ (𝐵m 𝐶))
Assertion
Ref Expression
unirnmapsn (𝜑𝑋 = (ran 𝑋m 𝐶))

Proof of Theorem unirnmapsn
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unirnmapsn.C . . . . 5 𝐶 = {𝐴}
2 snex 5295 . . . . 5 {𝐴} ∈ V
31, 2eqeltri 2829 . . . 4 𝐶 ∈ V
43a1i 11 . . 3 (𝜑𝐶 ∈ V)
5 unirnmapsn.x . . 3 (𝜑𝑋 ⊆ (𝐵m 𝐶))
64, 5unirnmap 42270 . 2 (𝜑𝑋 ⊆ (ran 𝑋m 𝐶))
7 simpl 486 . . . 4 ((𝜑𝑔 ∈ (ran 𝑋m 𝐶)) → 𝜑)
8 equid 2023 . . . . . . 7 𝑔 = 𝑔
9 rnuni 5975 . . . . . . . 8 ran 𝑋 = 𝑓𝑋 ran 𝑓
109oveq1i 7174 . . . . . . 7 (ran 𝑋m 𝐶) = ( 𝑓𝑋 ran 𝑓m 𝐶)
118, 10eleq12i 2825 . . . . . 6 (𝑔 ∈ (ran 𝑋m 𝐶) ↔ 𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶))
1211biimpi 219 . . . . 5 (𝑔 ∈ (ran 𝑋m 𝐶) → 𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶))
1312adantl 485 . . . 4 ((𝜑𝑔 ∈ (ran 𝑋m 𝐶)) → 𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶))
14 ovexd 7199 . . . . . . . . . 10 (𝜑 → (𝐵m 𝐶) ∈ V)
1514, 5ssexd 5189 . . . . . . . . 9 (𝜑𝑋 ∈ V)
16 rnexg 7628 . . . . . . . . . . 11 (𝑓𝑋 → ran 𝑓 ∈ V)
1716rgen 3063 . . . . . . . . . 10 𝑓𝑋 ran 𝑓 ∈ V
1817a1i 11 . . . . . . . . 9 (𝜑 → ∀𝑓𝑋 ran 𝑓 ∈ V)
19 iunexg 7682 . . . . . . . . 9 ((𝑋 ∈ V ∧ ∀𝑓𝑋 ran 𝑓 ∈ V) → 𝑓𝑋 ran 𝑓 ∈ V)
2015, 18, 19syl2anc 587 . . . . . . . 8 (𝜑 𝑓𝑋 ran 𝑓 ∈ V)
2120, 4elmapd 8444 . . . . . . 7 (𝜑 → (𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶) ↔ 𝑔:𝐶 𝑓𝑋 ran 𝑓))
2221biimpa 480 . . . . . 6 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶)) → 𝑔:𝐶 𝑓𝑋 ran 𝑓)
23 unirnmapsn.A . . . . . . . . 9 (𝜑𝐴𝑉)
24 snidg 4547 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ {𝐴})
2523, 24syl 17 . . . . . . . 8 (𝜑𝐴 ∈ {𝐴})
2625, 1eleqtrrdi 2844 . . . . . . 7 (𝜑𝐴𝐶)
2726adantr 484 . . . . . 6 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶)) → 𝐴𝐶)
2822, 27ffvelrnd 6856 . . . . 5 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶)) → (𝑔𝐴) ∈ 𝑓𝑋 ran 𝑓)
29 eliun 4882 . . . . 5 ((𝑔𝐴) ∈ 𝑓𝑋 ran 𝑓 ↔ ∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓)
3028, 29sylib 221 . . . 4 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶)) → ∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓)
317, 13, 30syl2anc 587 . . 3 ((𝜑𝑔 ∈ (ran 𝑋m 𝐶)) → ∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓)
32 elmapfn 8468 . . . . . 6 (𝑔 ∈ (ran 𝑋m 𝐶) → 𝑔 Fn 𝐶)
3332adantl 485 . . . . 5 ((𝜑𝑔 ∈ (ran 𝑋m 𝐶)) → 𝑔 Fn 𝐶)
34 simp3 1139 . . . . . . . . . . 11 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) ∈ ran 𝑓)
35233ad2ant1 1134 . . . . . . . . . . . 12 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝐴𝑉)
361oveq2i 7175 . . . . . . . . . . . . . . . . 17 (𝐵m 𝐶) = (𝐵m {𝐴})
375, 36sseqtrdi 3925 . . . . . . . . . . . . . . . 16 (𝜑𝑋 ⊆ (𝐵m {𝐴}))
3837adantr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑓𝑋) → 𝑋 ⊆ (𝐵m {𝐴}))
39 simpr 488 . . . . . . . . . . . . . . 15 ((𝜑𝑓𝑋) → 𝑓𝑋)
4038, 39sseldd 3876 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑋) → 𝑓 ∈ (𝐵m {𝐴}))
41 unirnmapsn.b . . . . . . . . . . . . . . . 16 (𝜑𝐵𝑊)
4241adantr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑓𝑋) → 𝐵𝑊)
432a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑓𝑋) → {𝐴} ∈ V)
4442, 43elmapd 8444 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑋) → (𝑓 ∈ (𝐵m {𝐴}) ↔ 𝑓:{𝐴}⟶𝐵))
4540, 44mpbid 235 . . . . . . . . . . . . 13 ((𝜑𝑓𝑋) → 𝑓:{𝐴}⟶𝐵)
46453adant3 1133 . . . . . . . . . . . 12 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓:{𝐴}⟶𝐵)
4735, 46rnsnf 42243 . . . . . . . . . . 11 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → ran 𝑓 = {(𝑓𝐴)})
4834, 47eleqtrd 2835 . . . . . . . . . 10 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) ∈ {(𝑓𝐴)})
49 fvex 6681 . . . . . . . . . . 11 (𝑔𝐴) ∈ V
5049elsn 4528 . . . . . . . . . 10 ((𝑔𝐴) ∈ {(𝑓𝐴)} ↔ (𝑔𝐴) = (𝑓𝐴))
5148, 50sylib 221 . . . . . . . . 9 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) = (𝑓𝐴))
52513adant1r 1178 . . . . . . . 8 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) = (𝑓𝐴))
5323adantr 484 . . . . . . . . . 10 ((𝜑𝑔 Fn 𝐶) → 𝐴𝑉)
54533ad2ant1 1134 . . . . . . . . 9 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝐴𝑉)
55 simp1r 1199 . . . . . . . . 9 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑔 Fn 𝐶)
5640, 36eleqtrrdi 2844 . . . . . . . . . . . 12 ((𝜑𝑓𝑋) → 𝑓 ∈ (𝐵m 𝐶))
57 elmapfn 8468 . . . . . . . . . . . 12 (𝑓 ∈ (𝐵m 𝐶) → 𝑓 Fn 𝐶)
5856, 57syl 17 . . . . . . . . . . 11 ((𝜑𝑓𝑋) → 𝑓 Fn 𝐶)
5958adantlr 715 . . . . . . . . . 10 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋) → 𝑓 Fn 𝐶)
60593adant3 1133 . . . . . . . . 9 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓 Fn 𝐶)
6154, 1, 55, 60fsneq 42268 . . . . . . . 8 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔 = 𝑓 ↔ (𝑔𝐴) = (𝑓𝐴)))
6252, 61mpbird 260 . . . . . . 7 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑔 = 𝑓)
63 simp2 1138 . . . . . . 7 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓𝑋)
6462, 63eqeltrd 2833 . . . . . 6 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑔𝑋)
65643exp 1120 . . . . 5 ((𝜑𝑔 Fn 𝐶) → (𝑓𝑋 → ((𝑔𝐴) ∈ ran 𝑓𝑔𝑋)))
667, 33, 65syl2anc 587 . . . 4 ((𝜑𝑔 ∈ (ran 𝑋m 𝐶)) → (𝑓𝑋 → ((𝑔𝐴) ∈ ran 𝑓𝑔𝑋)))
6766rexlimdv 3192 . . 3 ((𝜑𝑔 ∈ (ran 𝑋m 𝐶)) → (∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓𝑔𝑋))
6831, 67mpd 15 . 2 ((𝜑𝑔 ∈ (ran 𝑋m 𝐶)) → 𝑔𝑋)
696, 68eqelssd 3896 1 (𝜑𝑋 = (ran 𝑋m 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2113  wral 3053  wrex 3054  Vcvv 3397  wss 3841  {csn 4513   cuni 4793   ciun 4878  ran crn 5520   Fn wfn 6328  wf 6329  cfv 6333  (class class class)co 7164  m cmap 8430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-1st 7707  df-2nd 7708  df-map 8432
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator