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Theorem unirnmapsn 38880
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 4869 . . . . 5 {𝐴} ∈ V
31, 2eqeltri 2694 . . . 4 𝐶 ∈ V
43a1i 11 . . 3 (𝜑𝐶 ∈ V)
5 unirnmapsn.x . . 3 (𝜑𝑋 ⊆ (𝐵𝑚 𝐶))
64, 5unirnmap 38874 . 2 (𝜑𝑋 ⊆ (ran 𝑋𝑚 𝐶))
7 simpl 473 . . . . . 6 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → 𝜑)
8 equid 1936 . . . . . . . . 9 𝑔 = 𝑔
9 rnuni 5503 . . . . . . . . . 10 ran 𝑋 = 𝑓𝑋 ran 𝑓
109oveq1i 6614 . . . . . . . . 9 (ran 𝑋𝑚 𝐶) = ( 𝑓𝑋 ran 𝑓𝑚 𝐶)
118, 10eleq12i 2691 . . . . . . . 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 ovex 6632 . . . . . . . . . . . . 13 (𝐵𝑚 𝐶) ∈ V
1514a1i 11 . . . . . . . . . . . 12 (𝜑 → (𝐵𝑚 𝐶) ∈ V)
1615, 5ssexd 4765 . . . . . . . . . . 11 (𝜑𝑋 ∈ V)
17 rnexg 7045 . . . . . . . . . . . . 13 (𝑓𝑋 → ran 𝑓 ∈ V)
1817rgen 2917 . . . . . . . . . . . 12 𝑓𝑋 ran 𝑓 ∈ V
1918a1i 11 . . . . . . . . . . 11 (𝜑 → ∀𝑓𝑋 ran 𝑓 ∈ V)
20 iunexg 7089 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ ∀𝑓𝑋 ran 𝑓 ∈ V) → 𝑓𝑋 ran 𝑓 ∈ V)
2116, 19, 20syl2anc 692 . . . . . . . . . 10 (𝜑 𝑓𝑋 ran 𝑓 ∈ V)
2221, 4elmapd 7816 . . . . . . . . 9 (𝜑 → (𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶) ↔ 𝑔:𝐶 𝑓𝑋 ran 𝑓))
2322biimpa 501 . . . . . . . 8 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶)) → 𝑔:𝐶 𝑓𝑋 ran 𝑓)
24 unirnmapsn.A . . . . . . . . . . 11 (𝜑𝐴𝑉)
25 snidg 4177 . . . . . . . . . . 11 (𝐴𝑉𝐴 ∈ {𝐴})
2624, 25syl 17 . . . . . . . . . 10 (𝜑𝐴 ∈ {𝐴})
2726, 1syl6eleqr 2709 . . . . . . . . 9 (𝜑𝐴𝐶)
2827adantr 481 . . . . . . . 8 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶)) → 𝐴𝐶)
2923, 28ffvelrnd 6316 . . . . . . 7 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶)) → (𝑔𝐴) ∈ 𝑓𝑋 ran 𝑓)
30 eliun 4490 . . . . . . 7 ((𝑔𝐴) ∈ 𝑓𝑋 ran 𝑓 ↔ ∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓)
3129, 30sylib 208 . . . . . 6 ((𝜑𝑔 ∈ ( 𝑓𝑋 ran 𝑓𝑚 𝐶)) → ∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓)
327, 13, 31syl2anc 692 . . . . 5 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → ∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓)
33 elmapfn 7824 . . . . . . . 8 (𝑔 ∈ (ran 𝑋𝑚 𝐶) → 𝑔 Fn 𝐶)
3433adantl 482 . . . . . . 7 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → 𝑔 Fn 𝐶)
35 simp3 1061 . . . . . . . . . . . . 13 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) ∈ ran 𝑓)
36243ad2ant1 1080 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝐴𝑉)
371oveq2i 6615 . . . . . . . . . . . . . . . . . . 19 (𝐵𝑚 𝐶) = (𝐵𝑚 {𝐴})
385, 37syl6sseq 3630 . . . . . . . . . . . . . . . . . 18 (𝜑𝑋 ⊆ (𝐵𝑚 {𝐴}))
3938adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → 𝑋 ⊆ (𝐵𝑚 {𝐴}))
40 simpr 477 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → 𝑓𝑋)
4139, 40sseldd 3584 . . . . . . . . . . . . . . . 16 ((𝜑𝑓𝑋) → 𝑓 ∈ (𝐵𝑚 {𝐴}))
42 unirnmapsn.b . . . . . . . . . . . . . . . . . 18 (𝜑𝐵𝑊)
4342adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → 𝐵𝑊)
442a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → {𝐴} ∈ V)
4543, 44elmapd 7816 . . . . . . . . . . . . . . . 16 ((𝜑𝑓𝑋) → (𝑓 ∈ (𝐵𝑚 {𝐴}) ↔ 𝑓:{𝐴}⟶𝐵))
4641, 45mpbid 222 . . . . . . . . . . . . . . 15 ((𝜑𝑓𝑋) → 𝑓:{𝐴}⟶𝐵)
47463adant3 1079 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓:{𝐴}⟶𝐵)
4836, 47rnsnf 38844 . . . . . . . . . . . . 13 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → ran 𝑓 = {(𝑓𝐴)})
4935, 48eleqtrd 2700 . . . . . . . . . . . 12 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) ∈ {(𝑓𝐴)})
50 fvex 6158 . . . . . . . . . . . . 13 (𝑔𝐴) ∈ V
5150elsn 4163 . . . . . . . . . . . 12 ((𝑔𝐴) ∈ {(𝑓𝐴)} ↔ (𝑔𝐴) = (𝑓𝐴))
5249, 51sylib 208 . . . . . . . . . . 11 ((𝜑𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) = (𝑓𝐴))
53523adant1r 1316 . . . . . . . . . 10 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔𝐴) = (𝑓𝐴))
5424adantr 481 . . . . . . . . . . . 12 ((𝜑𝑔 Fn 𝐶) → 𝐴𝑉)
55543ad2ant1 1080 . . . . . . . . . . 11 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝐴𝑉)
56 simp1r 1084 . . . . . . . . . . 11 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑔 Fn 𝐶)
5741, 37syl6eleqr 2709 . . . . . . . . . . . . . 14 ((𝜑𝑓𝑋) → 𝑓 ∈ (𝐵𝑚 𝐶))
58 elmapfn 7824 . . . . . . . . . . . . . 14 (𝑓 ∈ (𝐵𝑚 𝐶) → 𝑓 Fn 𝐶)
5957, 58syl 17 . . . . . . . . . . . . 13 ((𝜑𝑓𝑋) → 𝑓 Fn 𝐶)
6059adantlr 750 . . . . . . . . . . . 12 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋) → 𝑓 Fn 𝐶)
61603adant3 1079 . . . . . . . . . . 11 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓 Fn 𝐶)
6255, 1, 56, 61fsneq 38872 . . . . . . . . . 10 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → (𝑔 = 𝑓 ↔ (𝑔𝐴) = (𝑓𝐴)))
6353, 62mpbird 247 . . . . . . . . 9 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑔 = 𝑓)
64 simp2 1060 . . . . . . . . 9 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑓𝑋)
6563, 64eqeltrd 2698 . . . . . . . 8 (((𝜑𝑔 Fn 𝐶) ∧ 𝑓𝑋 ∧ (𝑔𝐴) ∈ ran 𝑓) → 𝑔𝑋)
66653exp 1261 . . . . . . 7 ((𝜑𝑔 Fn 𝐶) → (𝑓𝑋 → ((𝑔𝐴) ∈ ran 𝑓𝑔𝑋)))
677, 34, 66syl2anc 692 . . . . . 6 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → (𝑓𝑋 → ((𝑔𝐴) ∈ ran 𝑓𝑔𝑋)))
6867rexlimdv 3023 . . . . 5 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → (∃𝑓𝑋 (𝑔𝐴) ∈ ran 𝑓𝑔𝑋))
6932, 68mpd 15 . . . 4 ((𝜑𝑔 ∈ (ran 𝑋𝑚 𝐶)) → 𝑔𝑋)
7069ralrimiva 2960 . . 3 (𝜑 → ∀𝑔 ∈ (ran 𝑋𝑚 𝐶)𝑔𝑋)
71 dfss3 3573 . . 3 ((ran 𝑋𝑚 𝐶) ⊆ 𝑋 ↔ ∀𝑔 ∈ (ran 𝑋𝑚 𝐶)𝑔𝑋)
7270, 71sylibr 224 . 2 (𝜑 → (ran 𝑋𝑚 𝐶) ⊆ 𝑋)
736, 72eqssd 3600 1 (𝜑𝑋 = (ran 𝑋𝑚 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  wss 3555  {csn 4148   cuni 4402   ciun 4485  ran crn 5075   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  𝑚 cmap 7802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804
This theorem is referenced by: (None)
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