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Theorem unirnmapsn 39874
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 5045 . . . . 5 {𝐴} ∈ V
31, 2eqeltri 2823 . . . 4 𝐶 ∈ V
43a1i 11 . . 3 (𝜑𝐶 ∈ V)
5 unirnmapsn.x . . 3 (𝜑𝑋 ⊆ (𝐵𝑚 𝐶))
64, 5unirnmap 39868 . 2 (𝜑𝑋 ⊆ (ran 𝑋𝑚 𝐶))
7 simpl 474 . . . . . 6 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → 𝜑)
8 equid 2082 . . . . . . . . 9 𝑔 = 𝑔
9 rnuni 5690 . . . . . . . . . 10 ran 𝑋 = 𝑓𝑋 ran 𝑓
109oveq1i 6811 . . . . . . . . 9 (ran 𝑋𝑚 𝐶) = ( 𝑓𝑋 ran 𝑓𝑚 𝐶)
118, 10eleq12i 2820 . . . . . . . 8 (𝑔 ∈ (ran 𝑋𝑚 𝐶) ↔ 𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶))
1211biimpi 206 . . . . . . 7 (𝑔 ∈ (ran 𝑋𝑚 𝐶) → 𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶))
1312adantl 473 . . . . . 6 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → 𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶))
14 ovexd 6831 . . . . . . . . . . . 12 (𝜑 → (𝐵𝑚 𝐶) ∈ V)
1514, 5ssexd 4945 . . . . . . . . . . 11 (𝜑𝑋 ∈ V)
16 rnexg 7251 . . . . . . . . . . . . 13 (𝑓𝑋 → ran 𝑓 ∈ V)
1716rgen 3048 . . . . . . . . . . . 12 𝑓𝑋 ran 𝑓 ∈ V
1817a1i 11 . . . . . . . . . . 11 (𝜑 → ∀𝑓𝑋 ran 𝑓 ∈ V)
19 iunexg 7296 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ ∀𝑓𝑋 ran 𝑓 ∈ V) → 𝑓𝑋 ran 𝑓 ∈ V)
2015, 18, 19syl2anc 696 . . . . . . . . . 10 (𝜑 𝑓𝑋 ran 𝑓 ∈ V)
2120, 4elmapd 8025 . . . . . . . . 9 (𝜑 → (𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶) ↔ 𝑔:𝐶 𝑓𝑋 ran 𝑓))
2221biimpa 502 . . . . . . . 8 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶)) → 𝑔:𝐶 𝑓𝑋 ran 𝑓)
23 unirnmapsn.A . . . . . . . . . . 11 (𝜑𝐴𝑉)
24 snidg 4339 . . . . . . . . . . 11 (𝐴𝑉𝐴 ∈ {𝐴})
2523, 24syl 17 . . . . . . . . . 10 (𝜑𝐴 ∈ {𝐴})
2625, 1syl6eleqr 2838 . . . . . . . . 9 (𝜑𝐴𝐶)
2726adantr 472 . . . . . . . 8 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶)) → 𝐴𝐶)
2822, 27ffvelrnd 6511 . . . . . . 7 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶)) → (𝑔𝐴) ∈ 𝑓𝑋 ran 𝑓)
29 eliun 4664 . . . . . . 7 ((𝑔𝐴) ∈ 𝑓𝑋 ran 𝑓 ↔ ∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓)
3028, 29sylib 208 . . . . . 6 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶)) → ∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓)
317, 13, 30syl2anc 696 . . . . 5 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → ∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓)
32 elmapfn 8034 . . . . . . . 8 (𝑔 ∈ (ran 𝑋𝑚 𝐶) → 𝑔 Fn 𝐶)
3332adantl 473 . . . . . . 7 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → 𝑔 Fn 𝐶)
34 simp3 1130 . . . . . . . . . . . . 13 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) ∈ ran 𝑓)
35233ad2ant1 1125 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝐴𝑉)
361oveq2i 6812 . . . . . . . . . . . . . . . . . . 19 (𝐵𝑚 𝐶) = (𝐵𝑚 {𝐴})
375, 36syl6sseq 3780 . . . . . . . . . . . . . . . . . 18 (𝜑𝑋 ⊆ (𝐵𝑚 {𝐴}))
3837adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → 𝑋 ⊆ (𝐵𝑚 {𝐴}))
39 simpr 479 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → 𝑓𝑋)
4038, 39sseldd 3733 . . . . . . . . . . . . . . . 16 ((𝜑𝑓𝑋) → 𝑓 ∈ (𝐵𝑚 {𝐴}))
41 unirnmapsn.b . . . . . . . . . . . . . . . . . 18 (𝜑𝐵𝑊)
4241adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → 𝐵𝑊)
432a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → {𝐴} ∈ V)
4442, 43elmapd 8025 . . . . . . . . . . . . . . . 16 ((𝜑𝑓𝑋) → (𝑓 ∈ (𝐵𝑚 {𝐴}) ↔ 𝑓:{𝐴}⟶𝐵))
4540, 44mpbid 222 . . . . . . . . . . . . . . 15 ((𝜑𝑓𝑋) → 𝑓:{𝐴}⟶𝐵)
46453adant3 1124 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓:{𝐴}⟶𝐵)
4735, 46rnsnf 39838 . . . . . . . . . . . . 13 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → ran 𝑓 = {(𝑓𝐴)})
4834, 47eleqtrd 2829 . . . . . . . . . . . 12 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) ∈ {(𝑓𝐴)})
49 fvex 6350 . . . . . . . . . . . . 13 (𝑔𝐴) ∈ V
5049elsn 4324 . . . . . . . . . . . 12 ((𝑔𝐴) ∈ {(𝑓𝐴)} ↔ (𝑔𝐴) = (𝑓𝐴))
5148, 50sylib 208 . . . . . . . . . . 11 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) = (𝑓𝐴))
52513adant1r 1168 . . . . . . . . . 10 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) = (𝑓𝐴))
5323adantr 472 . . . . . . . . . . . 12 ((𝜑𝑔 Fn 𝐶) → 𝐴𝑉)
54533ad2ant1 1125 . . . . . . . . . . 11 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝐴𝑉)
55 simp1r 1217 . . . . . . . . . . 11 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑔 Fn 𝐶)
5640, 36syl6eleqr 2838 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑋) → 𝑓 ∈ (𝐵𝑚 𝐶))
57 elmapfn 8034 . . . . . . . . . . . . . 14 (𝑓 ∈ (𝐵𝑚 𝐶) → 𝑓 Fn 𝐶)
5856, 57syl 17 . . . . . . . . . . . . 13 ((𝜑𝑓𝑋) → 𝑓 Fn 𝐶)
5958adantlr 753 . . . . . . . . . . . 12 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋) → 𝑓 Fn 𝐶)
60593adant3 1124 . . . . . . . . . . 11 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓 Fn 𝐶)
6154, 1, 55, 60fsneq 39866 . . . . . . . . . 10 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔 = 𝑓 ↔ (𝑔𝐴) = (𝑓𝐴)))
6252, 61mpbird 247 . . . . . . . . 9 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑔 = 𝑓)
63 simp2 1129 . . . . . . . . 9 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓𝑋)
6462, 63eqeltrd 2827 . . . . . . . 8 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑔𝑋)
65643exp 1112 . . . . . . 7 ((𝜑𝑔 Fn 𝐶) → (𝑓𝑋 → ((𝑔𝐴) ∈ ran 𝑓𝑔𝑋)))
667, 33, 65syl2anc 696 . . . . . 6 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → (𝑓𝑋 → ((𝑔𝐴) ∈ ran 𝑓𝑔𝑋)))
6766rexlimdv 3156 . . . . 5 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → (∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓𝑔𝑋))
6831, 67mpd 15 . . . 4 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → 𝑔𝑋)
6968ralrimiva 3092 . . 3 (𝜑 → ∀𝑔 ∈ (ran 𝑋𝑚 𝐶)𝑔𝑋)
70 dfss3 3721 . . 3 ((ran 𝑋𝑚 𝐶) ⊆ 𝑋 ↔ ∀𝑔 ∈ (ran 𝑋𝑚 𝐶)𝑔𝑋)
7169, 70sylibr 224 . 2 (𝜑 → (ran 𝑋𝑚 𝐶) ⊆ 𝑋)
726, 71eqssd 3749 1 (𝜑𝑋 = (ran 𝑋𝑚 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1620  wcel 2127  wral 3038  wrex 3039  Vcvv 3328  wss 3703  {csn 4309   cuni 4576   ciun 4660  ran crn 5255   Fn wfn 6032  wf 6033  cfv 6037  (class class class)co 6801  𝑚 cmap 8011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-1st 7321  df-2nd 7322  df-map 8013
This theorem is referenced by: (None)
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