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Theorem unirnmapsn 45219
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 5436 . . . . 5 {𝐴} ∈ V
31, 2eqeltri 2837 . . . 4 𝐶 ∈ V
43a1i 11 . . 3 (𝜑𝐶 ∈ V)
5 unirnmapsn.x . . 3 (𝜑𝑋 ⊆ (𝐵m 𝐶))
64, 5unirnmap 45213 . 2 (𝜑𝑋 ⊆ (ran 𝑋m 𝐶))
7 simpl 482 . . . 4 ((𝜑𝑔 ∈ (ran 𝑋m 𝐶)) → 𝜑)
8 equid 2011 . . . . . . 7 𝑔 = 𝑔
9 rnuni 6168 . . . . . . . 8 ran 𝑋 = 𝑓𝑋 ran 𝑓
109oveq1i 7441 . . . . . . 7 (ran 𝑋m 𝐶) = ( 𝑓𝑋 ran 𝑓m 𝐶)
118, 10eleq12i 2834 . . . . . 6 (𝑔 ∈ (ran 𝑋m 𝐶) ↔ 𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶))
1211biimpi 216 . . . . 5 (𝑔 ∈ (ran 𝑋m 𝐶) → 𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶))
1312adantl 481 . . . 4 ((𝜑𝑔 ∈ (ran 𝑋m 𝐶)) → 𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶))
14 ovexd 7466 . . . . . . . . . 10 (𝜑 → (𝐵m 𝐶) ∈ V)
1514, 5ssexd 5324 . . . . . . . . 9 (𝜑𝑋 ∈ V)
16 rnexg 7924 . . . . . . . . . . 11 (𝑓𝑋 → ran 𝑓 ∈ V)
1716rgen 3063 . . . . . . . . . 10 𝑓𝑋 ran 𝑓 ∈ V
1817a1i 11 . . . . . . . . 9 (𝜑 → ∀𝑓𝑋 ran 𝑓 ∈ V)
19 iunexg 7988 . . . . . . . . 9 ((𝑋 ∈ V ∧ ∀𝑓𝑋 ran 𝑓 ∈ V) → 𝑓𝑋 ran 𝑓 ∈ V)
2015, 18, 19syl2anc 584 . . . . . . . 8 (𝜑 𝑓𝑋 ran 𝑓 ∈ V)
2120, 4elmapd 8880 . . . . . . 7 (𝜑 → (𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶) ↔ 𝑔:𝐶 𝑓𝑋 ran 𝑓))
2221biimpa 476 . . . . . 6 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶)) → 𝑔:𝐶 𝑓𝑋 ran 𝑓)
23 unirnmapsn.A . . . . . . . . 9 (𝜑𝐴𝑉)
24 snidg 4660 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ {𝐴})
2523, 24syl 17 . . . . . . . 8 (𝜑𝐴 ∈ {𝐴})
2625, 1eleqtrrdi 2852 . . . . . . 7 (𝜑𝐴𝐶)
2726adantr 480 . . . . . 6 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶)) → 𝐴𝐶)
2822, 27ffvelcdmd 7105 . . . . 5 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶)) → (𝑔𝐴) ∈ 𝑓𝑋 ran 𝑓)
29 eliun 4995 . . . . 5 ((𝑔𝐴) ∈ 𝑓𝑋 ran 𝑓 ↔ ∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓)
3028, 29sylib 218 . . . 4 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶)) → ∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓)
317, 13, 30syl2anc 584 . . 3 ((𝜑𝑔 ∈ (ran 𝑋m 𝐶)) → ∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓)
32 elmapfn 8905 . . . . . 6 (𝑔 ∈ (ran 𝑋m 𝐶) → 𝑔 Fn 𝐶)
3332adantl 481 . . . . 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 7442 . . . . . . . . . . . . . . . . 17 (𝐵m 𝐶) = (𝐵m {𝐴})
375, 36sseqtrdi 4024 . . . . . . . . . . . . . . . 16 (𝜑𝑋 ⊆ (𝐵m {𝐴}))
3837adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑓𝑋) → 𝑋 ⊆ (𝐵m {𝐴}))
39 simpr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑓𝑋) → 𝑓𝑋)
4038, 39sseldd 3984 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑋) → 𝑓 ∈ (𝐵m {𝐴}))
41 unirnmapsn.b . . . . . . . . . . . . . . . 16 (𝜑𝐵𝑊)
4241adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑓𝑋) → 𝐵𝑊)
432a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑓𝑋) → {𝐴} ∈ V)
4442, 43elmapd 8880 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑋) → (𝑓 ∈ (𝐵m {𝐴}) ↔ 𝑓:{𝐴}⟶𝐵))
4540, 44mpbid 232 . . . . . . . . . . . . 13 ((𝜑𝑓𝑋) → 𝑓:{𝐴}⟶𝐵)
46453adant3 1133 . . . . . . . . . . . 12 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓:{𝐴}⟶𝐵)
4735, 46rnsnf 45189 . . . . . . . . . . 11 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → ran 𝑓 = {(𝑓𝐴)})
4834, 47eleqtrd 2843 . . . . . . . . . 10 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) ∈ {(𝑓𝐴)})
49 fvex 6919 . . . . . . . . . . 11 (𝑔𝐴) ∈ V
5049elsn 4641 . . . . . . . . . 10 ((𝑔𝐴) ∈ {(𝑓𝐴)} ↔ (𝑔𝐴) = (𝑓𝐴))
5148, 50sylib 218 . . . . . . . . 9 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) = (𝑓𝐴))
52513adant1r 1178 . . . . . . . 8 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) = (𝑓𝐴))
5323adantr 480 . . . . . . . . . 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 2852 . . . . . . . . . . . 12 ((𝜑𝑓𝑋) → 𝑓 ∈ (𝐵m 𝐶))
57 elmapfn 8905 . . . . . . . . . . . 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 45211 . . . . . . . 8 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔 = 𝑓 ↔ (𝑔𝐴) = (𝑓𝐴)))
6252, 61mpbird 257 . . . . . . 7 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑔 = 𝑓)
63 simp2 1138 . . . . . . 7 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓𝑋)
6462, 63eqeltrd 2841 . . . . . 6 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑔𝑋)
65643exp 1120 . . . . 5 ((𝜑𝑔 Fn 𝐶) → (𝑓𝑋 → ((𝑔𝐴) ∈ ran 𝑓𝑔𝑋)))
667, 33, 65syl2anc 584 . . . 4 ((𝜑𝑔 ∈ (ran 𝑋m 𝐶)) → (𝑓𝑋 → ((𝑔𝐴) ∈ ran 𝑓𝑔𝑋)))
6766rexlimdv 3153 . . 3 ((𝜑𝑔 ∈ (ran 𝑋m 𝐶)) → (∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓𝑔𝑋))
6831, 67mpd 15 . 2 ((𝜑𝑔 ∈ (ran 𝑋m 𝐶)) → 𝑔𝑋)
696, 68eqelssd 4005 1 (𝜑𝑋 = (ran 𝑋m 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  wss 3951  {csn 4626   cuni 4907   ciun 4991  ran crn 5686   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868
This theorem is referenced by: (None)
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