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Theorem unirnmapsn 42754
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 5354 . . . . 5 {𝐴} ∈ V
31, 2eqeltri 2835 . . . 4 𝐶 ∈ V
43a1i 11 . . 3 (𝜑𝐶 ∈ V)
5 unirnmapsn.x . . 3 (𝜑𝑋 ⊆ (𝐵m 𝐶))
64, 5unirnmap 42748 . 2 (𝜑𝑋 ⊆ (ran 𝑋m 𝐶))
7 simpl 483 . . . 4 ((𝜑𝑔 ∈ (ran 𝑋m 𝐶)) → 𝜑)
8 equid 2015 . . . . . . 7 𝑔 = 𝑔
9 rnuni 6052 . . . . . . . 8 ran 𝑋 = 𝑓𝑋 ran 𝑓
109oveq1i 7285 . . . . . . 7 (ran 𝑋m 𝐶) = ( 𝑓𝑋 ran 𝑓m 𝐶)
118, 10eleq12i 2831 . . . . . 6 (𝑔 ∈ (ran 𝑋m 𝐶) ↔ 𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶))
1211biimpi 215 . . . . 5 (𝑔 ∈ (ran 𝑋m 𝐶) → 𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶))
1312adantl 482 . . . 4 ((𝜑𝑔 ∈ (ran 𝑋m 𝐶)) → 𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶))
14 ovexd 7310 . . . . . . . . . 10 (𝜑 → (𝐵m 𝐶) ∈ V)
1514, 5ssexd 5248 . . . . . . . . 9 (𝜑𝑋 ∈ V)
16 rnexg 7751 . . . . . . . . . . 11 (𝑓𝑋 → ran 𝑓 ∈ V)
1716rgen 3074 . . . . . . . . . 10 𝑓𝑋 ran 𝑓 ∈ V
1817a1i 11 . . . . . . . . 9 (𝜑 → ∀𝑓𝑋 ran 𝑓 ∈ V)
19 iunexg 7806 . . . . . . . . 9 ((𝑋 ∈ V ∧ ∀𝑓𝑋 ran 𝑓 ∈ V) → 𝑓𝑋 ran 𝑓 ∈ V)
2015, 18, 19syl2anc 584 . . . . . . . 8 (𝜑 𝑓𝑋 ran 𝑓 ∈ V)
2120, 4elmapd 8629 . . . . . . 7 (𝜑 → (𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶) ↔ 𝑔:𝐶 𝑓𝑋 ran 𝑓))
2221biimpa 477 . . . . . 6 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶)) → 𝑔:𝐶 𝑓𝑋 ran 𝑓)
23 unirnmapsn.A . . . . . . . . 9 (𝜑𝐴𝑉)
24 snidg 4595 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ {𝐴})
2523, 24syl 17 . . . . . . . 8 (𝜑𝐴 ∈ {𝐴})
2625, 1eleqtrrdi 2850 . . . . . . 7 (𝜑𝐴𝐶)
2726adantr 481 . . . . . 6 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶)) → 𝐴𝐶)
2822, 27ffvelrnd 6962 . . . . 5 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓m 𝐶)) → (𝑔𝐴) ∈ 𝑓𝑋 ran 𝑓)
29 eliun 4928 . . . . 5 ((𝑔𝐴) ∈ 𝑓𝑋 ran 𝑓 ↔ ∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓)
3028, 29sylib 217 . . . 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 8653 . . . . . 6 (𝑔 ∈ (ran 𝑋m 𝐶) → 𝑔 Fn 𝐶)
3332adantl 482 . . . . 5 ((𝜑𝑔 ∈ (ran 𝑋m 𝐶)) → 𝑔 Fn 𝐶)
34 simp3 1137 . . . . . . . . . . 11 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) ∈ ran 𝑓)
35233ad2ant1 1132 . . . . . . . . . . . 12 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝐴𝑉)
361oveq2i 7286 . . . . . . . . . . . . . . . . 17 (𝐵m 𝐶) = (𝐵m {𝐴})
375, 36sseqtrdi 3971 . . . . . . . . . . . . . . . 16 (𝜑𝑋 ⊆ (𝐵m {𝐴}))
3837adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑓𝑋) → 𝑋 ⊆ (𝐵m {𝐴}))
39 simpr 485 . . . . . . . . . . . . . . 15 ((𝜑𝑓𝑋) → 𝑓𝑋)
4038, 39sseldd 3922 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑋) → 𝑓 ∈ (𝐵m {𝐴}))
41 unirnmapsn.b . . . . . . . . . . . . . . . 16 (𝜑𝐵𝑊)
4241adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑓𝑋) → 𝐵𝑊)
432a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑓𝑋) → {𝐴} ∈ V)
4442, 43elmapd 8629 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑋) → (𝑓 ∈ (𝐵m {𝐴}) ↔ 𝑓:{𝐴}⟶𝐵))
4540, 44mpbid 231 . . . . . . . . . . . . 13 ((𝜑𝑓𝑋) → 𝑓:{𝐴}⟶𝐵)
46453adant3 1131 . . . . . . . . . . . 12 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓:{𝐴}⟶𝐵)
4735, 46rnsnf 42721 . . . . . . . . . . 11 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → ran 𝑓 = {(𝑓𝐴)})
4834, 47eleqtrd 2841 . . . . . . . . . 10 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) ∈ {(𝑓𝐴)})
49 fvex 6787 . . . . . . . . . . 11 (𝑔𝐴) ∈ V
5049elsn 4576 . . . . . . . . . 10 ((𝑔𝐴) ∈ {(𝑓𝐴)} ↔ (𝑔𝐴) = (𝑓𝐴))
5148, 50sylib 217 . . . . . . . . 9 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) = (𝑓𝐴))
52513adant1r 1176 . . . . . . . 8 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) = (𝑓𝐴))
5323adantr 481 . . . . . . . . . 10 ((𝜑𝑔 Fn 𝐶) → 𝐴𝑉)
54533ad2ant1 1132 . . . . . . . . 9 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝐴𝑉)
55 simp1r 1197 . . . . . . . . 9 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑔 Fn 𝐶)
5640, 36eleqtrrdi 2850 . . . . . . . . . . . 12 ((𝜑𝑓𝑋) → 𝑓 ∈ (𝐵m 𝐶))
57 elmapfn 8653 . . . . . . . . . . . 12 (𝑓 ∈ (𝐵m 𝐶) → 𝑓 Fn 𝐶)
5856, 57syl 17 . . . . . . . . . . 11 ((𝜑𝑓𝑋) → 𝑓 Fn 𝐶)
5958adantlr 712 . . . . . . . . . 10 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋) → 𝑓 Fn 𝐶)
60593adant3 1131 . . . . . . . . 9 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓 Fn 𝐶)
6154, 1, 55, 60fsneq 42746 . . . . . . . 8 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔 = 𝑓 ↔ (𝑔𝐴) = (𝑓𝐴)))
6252, 61mpbird 256 . . . . . . 7 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑔 = 𝑓)
63 simp2 1136 . . . . . . 7 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓𝑋)
6462, 63eqeltrd 2839 . . . . . 6 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑔𝑋)
65643exp 1118 . . . . 5 ((𝜑𝑔 Fn 𝐶) → (𝑓𝑋 → ((𝑔𝐴) ∈ ran 𝑓𝑔𝑋)))
667, 33, 65syl2anc 584 . . . 4 ((𝜑𝑔 ∈ (ran 𝑋m 𝐶)) → (𝑓𝑋 → ((𝑔𝐴) ∈ ran 𝑓𝑔𝑋)))
6766rexlimdv 3212 . . 3 ((𝜑𝑔 ∈ (ran 𝑋m 𝐶)) → (∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓𝑔𝑋))
6831, 67mpd 15 . 2 ((𝜑𝑔 ∈ (ran 𝑋m 𝐶)) → 𝑔𝑋)
696, 68eqelssd 3942 1 (𝜑𝑋 = (ran 𝑋m 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  wrex 3065  Vcvv 3432  wss 3887  {csn 4561   cuni 4839   ciun 4924  ran crn 5590   Fn wfn 6428  wf 6429  cfv 6433  (class class class)co 7275  m cmap 8615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617
This theorem is referenced by: (None)
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