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

Theorem unirnmapsn 38401
 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 4835 . . . . 5 {𝐴} ∈ V
31, 2eqeltri 2684 . . . 4 𝐶 ∈ V
43a1i 11 . . 3 (𝜑𝐶 ∈ V)
5 unirnmapsn.x . . 3 (𝜑𝑋 ⊆ (𝐵𝑚 𝐶))
64, 5unirnmap 38395 . 2 (𝜑𝑋 ⊆ (ran 𝑋𝑚 𝐶))
7 simpl 472 . . . . . 6 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → 𝜑)
8 equid 1926 . . . . . . . . 9 𝑔 = 𝑔
9 rnuni 5463 . . . . . . . . . 10 ran 𝑋 = 𝑓𝑋 ran 𝑓
109oveq1i 6559 . . . . . . . . 9 (ran 𝑋𝑚 𝐶) = ( 𝑓𝑋 ran 𝑓𝑚 𝐶)
118, 10eleq12i 2681 . . . . . . . 8 (𝑔 ∈ (ran 𝑋𝑚 𝐶) ↔ 𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶))
1211biimpi 205 . . . . . . 7 (𝑔 ∈ (ran 𝑋𝑚 𝐶) → 𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶))
1312adantl 481 . . . . . 6 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → 𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶))
14 ovex 6577 . . . . . . . . . . . . 13 (𝐵𝑚 𝐶) ∈ V
1514a1i 11 . . . . . . . . . . . 12 (𝜑 → (𝐵𝑚 𝐶) ∈ V)
1615, 5ssexd 4733 . . . . . . . . . . 11 (𝜑𝑋 ∈ V)
17 rnexg 6990 . . . . . . . . . . . . 13 (𝑓𝑋 → ran 𝑓 ∈ V)
1817rgen 2906 . . . . . . . . . . . 12 𝑓𝑋 ran 𝑓 ∈ V
1918a1i 11 . . . . . . . . . . 11 (𝜑 → ∀𝑓𝑋 ran 𝑓 ∈ V)
20 iunexg 7035 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ ∀𝑓𝑋 ran 𝑓 ∈ V) → 𝑓𝑋 ran 𝑓 ∈ V)
2116, 19, 20syl2anc 691 . . . . . . . . . 10 (𝜑 𝑓𝑋 ran 𝑓 ∈ V)
2221, 4elmapd 7758 . . . . . . . . 9 (𝜑 → (𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶) ↔ 𝑔:𝐶 𝑓𝑋 ran 𝑓))
2322biimpa 500 . . . . . . . 8 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶)) → 𝑔:𝐶 𝑓𝑋 ran 𝑓)
24 unirnmapsn.A . . . . . . . . . . 11 (𝜑𝐴𝑉)
25 snidg 4153 . . . . . . . . . . 11 (𝐴𝑉𝐴 ∈ {𝐴})
2624, 25syl 17 . . . . . . . . . 10 (𝜑𝐴 ∈ {𝐴})
2726, 1syl6eleqr 2699 . . . . . . . . 9 (𝜑𝐴𝐶)
2827adantr 480 . . . . . . . 8 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶)) → 𝐴𝐶)
2923, 28ffvelrnd 6268 . . . . . . 7 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶)) → (𝑔𝐴) ∈ 𝑓𝑋 ran 𝑓)
30 eliun 4460 . . . . . . 7 ((𝑔𝐴) ∈ 𝑓𝑋 ran 𝑓 ↔ ∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓)
3129, 30sylib 207 . . . . . 6 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶)) → ∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓)
327, 13, 31syl2anc 691 . . . . 5 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → ∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓)
33 elmapfn 7766 . . . . . . . 8 (𝑔 ∈ (ran 𝑋𝑚 𝐶) → 𝑔 Fn 𝐶)
3433adantl 481 . . . . . . 7 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → 𝑔 Fn 𝐶)
35 simp3 1056 . . . . . . . . . . . . 13 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) ∈ ran 𝑓)
36243ad2ant1 1075 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝐴𝑉)
371oveq2i 6560 . . . . . . . . . . . . . . . . . . 19 (𝐵𝑚 𝐶) = (𝐵𝑚 {𝐴})
385, 37syl6sseq 3614 . . . . . . . . . . . . . . . . . 18 (𝜑𝑋 ⊆ (𝐵𝑚 {𝐴}))
3938adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → 𝑋 ⊆ (𝐵𝑚 {𝐴}))
40 simpr 476 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → 𝑓𝑋)
4139, 40sseldd 3569 . . . . . . . . . . . . . . . 16 ((𝜑𝑓𝑋) → 𝑓 ∈ (𝐵𝑚 {𝐴}))
42 unirnmapsn.b . . . . . . . . . . . . . . . . . 18 (𝜑𝐵𝑊)
4342adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → 𝐵𝑊)
442a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → {𝐴} ∈ V)
4543, 44elmapd 7758 . . . . . . . . . . . . . . . 16 ((𝜑𝑓𝑋) → (𝑓 ∈ (𝐵𝑚 {𝐴}) ↔ 𝑓:{𝐴}⟶𝐵))
4641, 45mpbid 221 . . . . . . . . . . . . . . 15 ((𝜑𝑓𝑋) → 𝑓:{𝐴}⟶𝐵)
47463adant3 1074 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓:{𝐴}⟶𝐵)
4836, 47rnsnf 38365 . . . . . . . . . . . . 13 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → ran 𝑓 = {(𝑓𝐴)})
4935, 48eleqtrd 2690 . . . . . . . . . . . 12 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) ∈ {(𝑓𝐴)})
50 fvex 6113 . . . . . . . . . . . . 13 (𝑔𝐴) ∈ V
5150elsn 4140 . . . . . . . . . . . 12 ((𝑔𝐴) ∈ {(𝑓𝐴)} ↔ (𝑔𝐴) = (𝑓𝐴))
5249, 51sylib 207 . . . . . . . . . . 11 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) = (𝑓𝐴))
53523adant1r 1311 . . . . . . . . . 10 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) = (𝑓𝐴))
5424adantr 480 . . . . . . . . . . . 12 ((𝜑𝑔 Fn 𝐶) → 𝐴𝑉)
55543ad2ant1 1075 . . . . . . . . . . 11 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝐴𝑉)
56 simp1r 1079 . . . . . . . . . . 11 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑔 Fn 𝐶)
5741, 37syl6eleqr 2699 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑋) → 𝑓 ∈ (𝐵𝑚 𝐶))
58 elmapfn 7766 . . . . . . . . . . . . . 14 (𝑓 ∈ (𝐵𝑚 𝐶) → 𝑓 Fn 𝐶)
5957, 58syl 17 . . . . . . . . . . . . 13 ((𝜑𝑓𝑋) → 𝑓 Fn 𝐶)
6059adantlr 747 . . . . . . . . . . . 12 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋) → 𝑓 Fn 𝐶)
61603adant3 1074 . . . . . . . . . . 11 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓 Fn 𝐶)
6255, 1, 56, 61fsneq 38393 . . . . . . . . . 10 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔 = 𝑓 ↔ (𝑔𝐴) = (𝑓𝐴)))
6353, 62mpbird 246 . . . . . . . . 9 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑔 = 𝑓)
64 simp2 1055 . . . . . . . . 9 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓𝑋)
6563, 64eqeltrd 2688 . . . . . . . 8 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑔𝑋)
66653exp 1256 . . . . . . 7 ((𝜑𝑔 Fn 𝐶) → (𝑓𝑋 → ((𝑔𝐴) ∈ ran 𝑓𝑔𝑋)))
677, 34, 66syl2anc 691 . . . . . 6 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → (𝑓𝑋 → ((𝑔𝐴) ∈ ran 𝑓𝑔𝑋)))
6867rexlimdv 3012 . . . . 5 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → (∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓𝑔𝑋))
6932, 68mpd 15 . . . 4 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → 𝑔𝑋)
7069ralrimiva 2949 . . 3 (𝜑 → ∀𝑔 ∈ (ran 𝑋𝑚 𝐶)𝑔𝑋)
71 dfss3 3558 . . 3 ((ran 𝑋𝑚 𝐶) ⊆ 𝑋 ↔ ∀𝑔 ∈ (ran 𝑋𝑚 𝐶)𝑔𝑋)
7270, 71sylibr 223 . 2 (𝜑 → (ran 𝑋𝑚 𝐶) ⊆ 𝑋)
736, 72eqssd 3585 1 (𝜑𝑋 = (ran 𝑋𝑚 𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  {csn 4125  ∪ cuni 4372  ∪ ciun 4455  ran crn 5039   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746 This theorem is referenced by: (None)
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