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Theorem unirnmapsn 39923
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 5057 . . . . 5 {𝐴} ∈ V
31, 2eqeltri 2835 . . . 4 𝐶 ∈ V
43a1i 11 . . 3 (𝜑𝐶 ∈ V)
5 unirnmapsn.x . . 3 (𝜑𝑋 ⊆ (𝐵𝑚 𝐶))
64, 5unirnmap 39917 . 2 (𝜑𝑋 ⊆ (ran 𝑋𝑚 𝐶))
7 simpl 474 . . . . . 6 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → 𝜑)
8 equid 2094 . . . . . . . . 9 𝑔 = 𝑔
9 rnuni 5702 . . . . . . . . . 10 ran 𝑋 = 𝑓𝑋 ran 𝑓
109oveq1i 6824 . . . . . . . . 9 (ran 𝑋𝑚 𝐶) = ( 𝑓𝑋 ran 𝑓𝑚 𝐶)
118, 10eleq12i 2832 . . . . . . . 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 6844 . . . . . . . . . . . 12 (𝜑 → (𝐵𝑚 𝐶) ∈ V)
1514, 5ssexd 4957 . . . . . . . . . . 11 (𝜑𝑋 ∈ V)
16 rnexg 7264 . . . . . . . . . . . . 13 (𝑓𝑋 → ran 𝑓 ∈ V)
1716rgen 3060 . . . . . . . . . . . 12 𝑓𝑋 ran 𝑓 ∈ V
1817a1i 11 . . . . . . . . . . 11 (𝜑 → ∀𝑓𝑋 ran 𝑓 ∈ V)
19 iunexg 7309 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ ∀𝑓𝑋 ran 𝑓 ∈ V) → 𝑓𝑋 ran 𝑓 ∈ V)
2015, 18, 19syl2anc 696 . . . . . . . . . 10 (𝜑 𝑓𝑋 ran 𝑓 ∈ V)
2120, 4elmapd 8039 . . . . . . . . 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 4351 . . . . . . . . . . 11 (𝐴𝑉𝐴 ∈ {𝐴})
2523, 24syl 17 . . . . . . . . . 10 (𝜑𝐴 ∈ {𝐴})
2625, 1syl6eleqr 2850 . . . . . . . . 9 (𝜑𝐴𝐶)
2726adantr 472 . . . . . . . 8 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶)) → 𝐴𝐶)
2822, 27ffvelrnd 6524 . . . . . . 7 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶)) → (𝑔𝐴) ∈ 𝑓𝑋 ran 𝑓)
29 eliun 4676 . . . . . . 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 8048 . . . . . . . 8 (𝑔 ∈ (ran 𝑋𝑚 𝐶) → 𝑔 Fn 𝐶)
3332adantl 473 . . . . . . 7 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → 𝑔 Fn 𝐶)
34 simp3 1133 . . . . . . . . . . . . 13 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) ∈ ran 𝑓)
35233ad2ant1 1128 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝐴𝑉)
361oveq2i 6825 . . . . . . . . . . . . . . . . . . 19 (𝐵𝑚 𝐶) = (𝐵𝑚 {𝐴})
375, 36syl6sseq 3792 . . . . . . . . . . . . . . . . . 18 (𝜑𝑋 ⊆ (𝐵𝑚 {𝐴}))
3837adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → 𝑋 ⊆ (𝐵𝑚 {𝐴}))
39 simpr 479 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → 𝑓𝑋)
4038, 39sseldd 3745 . . . . . . . . . . . . . . . 16 ((𝜑𝑓𝑋) → 𝑓 ∈ (𝐵𝑚 {𝐴}))
41 unirnmapsn.b . . . . . . . . . . . . . . . . . 18 (𝜑𝐵𝑊)
4241adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → 𝐵𝑊)
432a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → {𝐴} ∈ V)
4442, 43elmapd 8039 . . . . . . . . . . . . . . . 16 ((𝜑𝑓𝑋) → (𝑓 ∈ (𝐵𝑚 {𝐴}) ↔ 𝑓:{𝐴}⟶𝐵))
4540, 44mpbid 222 . . . . . . . . . . . . . . 15 ((𝜑𝑓𝑋) → 𝑓:{𝐴}⟶𝐵)
46453adant3 1127 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓:{𝐴}⟶𝐵)
4735, 46rnsnf 39887 . . . . . . . . . . . . 13 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → ran 𝑓 = {(𝑓𝐴)})
4834, 47eleqtrd 2841 . . . . . . . . . . . 12 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) ∈ {(𝑓𝐴)})
49 fvex 6363 . . . . . . . . . . . . 13 (𝑔𝐴) ∈ V
5049elsn 4336 . . . . . . . . . . . 12 ((𝑔𝐴) ∈ {(𝑓𝐴)} ↔ (𝑔𝐴) = (𝑓𝐴))
5148, 50sylib 208 . . . . . . . . . . 11 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) = (𝑓𝐴))
52513adant1r 1188 . . . . . . . . . 10 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) = (𝑓𝐴))
5323adantr 472 . . . . . . . . . . . 12 ((𝜑𝑔 Fn 𝐶) → 𝐴𝑉)
54533ad2ant1 1128 . . . . . . . . . . 11 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝐴𝑉)
55 simp1r 1241 . . . . . . . . . . 11 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑔 Fn 𝐶)
5640, 36syl6eleqr 2850 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑋) → 𝑓 ∈ (𝐵𝑚 𝐶))
57 elmapfn 8048 . . . . . . . . . . . . . 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 1127 . . . . . . . . . . 11 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓 Fn 𝐶)
6154, 1, 55, 60fsneq 39915 . . . . . . . . . 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 1132 . . . . . . . . 9 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓𝑋)
6462, 63eqeltrd 2839 . . . . . . . 8 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑔𝑋)
65643exp 1113 . . . . . . 7 ((𝜑𝑔 Fn 𝐶) → (𝑓𝑋 → ((𝑔𝐴) ∈ ran 𝑓𝑔𝑋)))
667, 33, 65syl2anc 696 . . . . . 6 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → (𝑓𝑋 → ((𝑔𝐴) ∈ ran 𝑓𝑔𝑋)))
6766rexlimdv 3168 . . . . 5 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → (∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓𝑔𝑋))
6831, 67mpd 15 . . . 4 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → 𝑔𝑋)
6968ralrimiva 3104 . . 3 (𝜑 → ∀𝑔 ∈ (ran 𝑋𝑚 𝐶)𝑔𝑋)
70 dfss3 3733 . . 3 ((ran 𝑋𝑚 𝐶) ⊆ 𝑋 ↔ ∀𝑔 ∈ (ran 𝑋𝑚 𝐶)𝑔𝑋)
7169, 70sylibr 224 . 2 (𝜑 → (ran 𝑋𝑚 𝐶) ⊆ 𝑋)
726, 71eqssd 3761 1 (𝜑𝑋 = (ran 𝑋𝑚 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  wss 3715  {csn 4321   cuni 4588   ciun 4672  ran crn 5267   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6814  𝑚 cmap 8025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-map 8027
This theorem is referenced by: (None)
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