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Theorem unirnmapsn 39222
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 4899 . . . . 5 {𝐴} ∈ V
31, 2eqeltri 2695 . . . 4 𝐶 ∈ V
43a1i 11 . . 3 (𝜑𝐶 ∈ V)
5 unirnmapsn.x . . 3 (𝜑𝑋 ⊆ (𝐵𝑚 𝐶))
64, 5unirnmap 39216 . 2 (𝜑𝑋 ⊆ (ran 𝑋𝑚 𝐶))
7 simpl 473 . . . . . 6 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → 𝜑)
8 equid 1937 . . . . . . . . 9 𝑔 = 𝑔
9 rnuni 5532 . . . . . . . . . 10 ran 𝑋 = 𝑓𝑋 ran 𝑓
109oveq1i 6645 . . . . . . . . 9 (ran 𝑋𝑚 𝐶) = ( 𝑓𝑋 ran 𝑓𝑚 𝐶)
118, 10eleq12i 2692 . . . . . . . 8 (𝑔 ∈ (ran 𝑋𝑚 𝐶) ↔ 𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶))
1211biimpi 206 . . . . . . 7 (𝑔 ∈ (ran 𝑋𝑚 𝐶) → 𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶))
1312adantl 482 . . . . . 6 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → 𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶))
14 ovexd 6665 . . . . . . . . . . . 12 (𝜑 → (𝐵𝑚 𝐶) ∈ V)
1514, 5ssexd 4796 . . . . . . . . . . 11 (𝜑𝑋 ∈ V)
16 rnexg 7083 . . . . . . . . . . . . 13 (𝑓𝑋 → ran 𝑓 ∈ V)
1716rgen 2919 . . . . . . . . . . . 12 𝑓𝑋 ran 𝑓 ∈ V
1817a1i 11 . . . . . . . . . . 11 (𝜑 → ∀𝑓𝑋 ran 𝑓 ∈ V)
19 iunexg 7128 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ ∀𝑓𝑋 ran 𝑓 ∈ V) → 𝑓𝑋 ran 𝑓 ∈ V)
2015, 18, 19syl2anc 692 . . . . . . . . . 10 (𝜑 𝑓𝑋 ran 𝑓 ∈ V)
2120, 4elmapd 7856 . . . . . . . . 9 (𝜑 → (𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶) ↔ 𝑔:𝐶 𝑓𝑋 ran 𝑓))
2221biimpa 501 . . . . . . . 8 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶)) → 𝑔:𝐶 𝑓𝑋 ran 𝑓)
23 unirnmapsn.A . . . . . . . . . . 11 (𝜑𝐴𝑉)
24 snidg 4197 . . . . . . . . . . 11 (𝐴𝑉𝐴 ∈ {𝐴})
2523, 24syl 17 . . . . . . . . . 10 (𝜑𝐴 ∈ {𝐴})
2625, 1syl6eleqr 2710 . . . . . . . . 9 (𝜑𝐴𝐶)
2726adantr 481 . . . . . . . 8 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶)) → 𝐴𝐶)
2822, 27ffvelrnd 6346 . . . . . . 7 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶)) → (𝑔𝐴) ∈ 𝑓𝑋 ran 𝑓)
29 eliun 4515 . . . . . . 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 692 . . . . 5 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → ∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓)
32 elmapfn 7865 . . . . . . . 8 (𝑔 ∈ (ran 𝑋𝑚 𝐶) → 𝑔 Fn 𝐶)
3332adantl 482 . . . . . . 7 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → 𝑔 Fn 𝐶)
34 simp3 1061 . . . . . . . . . . . . 13 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) ∈ ran 𝑓)
35233ad2ant1 1080 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝐴𝑉)
361oveq2i 6646 . . . . . . . . . . . . . . . . . . 19 (𝐵𝑚 𝐶) = (𝐵𝑚 {𝐴})
375, 36syl6sseq 3643 . . . . . . . . . . . . . . . . . 18 (𝜑𝑋 ⊆ (𝐵𝑚 {𝐴}))
3837adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → 𝑋 ⊆ (𝐵𝑚 {𝐴}))
39 simpr 477 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → 𝑓𝑋)
4038, 39sseldd 3596 . . . . . . . . . . . . . . . 16 ((𝜑𝑓𝑋) → 𝑓 ∈ (𝐵𝑚 {𝐴}))
41 unirnmapsn.b . . . . . . . . . . . . . . . . . 18 (𝜑𝐵𝑊)
4241adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → 𝐵𝑊)
432a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → {𝐴} ∈ V)
4442, 43elmapd 7856 . . . . . . . . . . . . . . . 16 ((𝜑𝑓𝑋) → (𝑓 ∈ (𝐵𝑚 {𝐴}) ↔ 𝑓:{𝐴}⟶𝐵))
4540, 44mpbid 222 . . . . . . . . . . . . . . 15 ((𝜑𝑓𝑋) → 𝑓:{𝐴}⟶𝐵)
46453adant3 1079 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓:{𝐴}⟶𝐵)
4735, 46rnsnf 39186 . . . . . . . . . . . . 13 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → ran 𝑓 = {(𝑓𝐴)})
4834, 47eleqtrd 2701 . . . . . . . . . . . 12 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) ∈ {(𝑓𝐴)})
49 fvex 6188 . . . . . . . . . . . . 13 (𝑔𝐴) ∈ V
5049elsn 4183 . . . . . . . . . . . 12 ((𝑔𝐴) ∈ {(𝑓𝐴)} ↔ (𝑔𝐴) = (𝑓𝐴))
5148, 50sylib 208 . . . . . . . . . . 11 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) = (𝑓𝐴))
52513adant1r 1317 . . . . . . . . . 10 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) = (𝑓𝐴))
5323adantr 481 . . . . . . . . . . . 12 ((𝜑𝑔 Fn 𝐶) → 𝐴𝑉)
54533ad2ant1 1080 . . . . . . . . . . 11 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝐴𝑉)
55 simp1r 1084 . . . . . . . . . . 11 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑔 Fn 𝐶)
5640, 36syl6eleqr 2710 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑋) → 𝑓 ∈ (𝐵𝑚 𝐶))
57 elmapfn 7865 . . . . . . . . . . . . . 14 (𝑓 ∈ (𝐵𝑚 𝐶) → 𝑓 Fn 𝐶)
5856, 57syl 17 . . . . . . . . . . . . 13 ((𝜑𝑓𝑋) → 𝑓 Fn 𝐶)
5958adantlr 750 . . . . . . . . . . . 12 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋) → 𝑓 Fn 𝐶)
60593adant3 1079 . . . . . . . . . . 11 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓 Fn 𝐶)
6154, 1, 55, 60fsneq 39214 . . . . . . . . . 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 1060 . . . . . . . . 9 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓𝑋)
6462, 63eqeltrd 2699 . . . . . . . 8 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑔𝑋)
65643exp 1262 . . . . . . 7 ((𝜑𝑔 Fn 𝐶) → (𝑓𝑋 → ((𝑔𝐴) ∈ ran 𝑓𝑔𝑋)))
667, 33, 65syl2anc 692 . . . . . 6 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → (𝑓𝑋 → ((𝑔𝐴) ∈ ran 𝑓𝑔𝑋)))
6766rexlimdv 3026 . . . . 5 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → (∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓𝑔𝑋))
6831, 67mpd 15 . . . 4 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → 𝑔𝑋)
6968ralrimiva 2963 . . 3 (𝜑 → ∀𝑔 ∈ (ran 𝑋𝑚 𝐶)𝑔𝑋)
70 dfss3 3585 . . 3 ((ran 𝑋𝑚 𝐶) ⊆ 𝑋 ↔ ∀𝑔 ∈ (ran 𝑋𝑚 𝐶)𝑔𝑋)
7169, 70sylibr 224 . 2 (𝜑 → (ran 𝑋𝑚 𝐶) ⊆ 𝑋)
726, 71eqssd 3612 1 (𝜑𝑋 = (ran 𝑋𝑚 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  wrex 2910  Vcvv 3195  wss 3567  {csn 4168   cuni 4427   ciun 4511  ran crn 5105   Fn wfn 5871  wf 5872  cfv 5876  (class class class)co 6635  𝑚 cmap 7842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-map 7844
This theorem is referenced by: (None)
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