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Theorem sbthlemi9 6912
Description: Lemma for isbth 6914. (Contributed by NM, 28-Mar-1998.)
Hypotheses
Ref Expression
sbthlem.1 𝐴 ∈ V
sbthlem.2 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
sbthlem.3 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
Assertion
Ref Expression
sbthlemi9 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝐻:𝐴1-1-onto𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝑓   𝑥,𝑔   𝑥,𝐻
Allowed substitution hints:   𝐴(𝑓,𝑔)   𝐵(𝑓,𝑔)   𝐷(𝑓,𝑔)   𝐻(𝑓,𝑔)

Proof of Theorem sbthlemi9
StepHypRef Expression
1 simp2 983 . . . . . . . . . 10 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝑓:𝐴1-1𝐵)
2 df-f1 5178 . . . . . . . . . 10 (𝑓:𝐴1-1𝐵 ↔ (𝑓:𝐴𝐵 ∧ Fun 𝑓))
31, 2sylib 121 . . . . . . . . 9 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → (𝑓:𝐴𝐵 ∧ Fun 𝑓))
43simpld 111 . . . . . . . 8 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝑓:𝐴𝐵)
5 df-f 5177 . . . . . . . 8 (𝑓:𝐴𝐵 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓𝐵))
64, 5sylib 121 . . . . . . 7 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → (𝑓 Fn 𝐴 ∧ ran 𝑓𝐵))
76simpld 111 . . . . . 6 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝑓 Fn 𝐴)
8 df-fn 5176 . . . . . 6 (𝑓 Fn 𝐴 ↔ (Fun 𝑓 ∧ dom 𝑓 = 𝐴))
97, 8sylib 121 . . . . 5 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → (Fun 𝑓 ∧ dom 𝑓 = 𝐴))
109simpld 111 . . . 4 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → Fun 𝑓)
11 simp3 984 . . . . . 6 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝑔:𝐵1-1𝐴)
12 df-f1 5178 . . . . . 6 (𝑔:𝐵1-1𝐴 ↔ (𝑔:𝐵𝐴 ∧ Fun 𝑔))
1311, 12sylib 121 . . . . 5 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → (𝑔:𝐵𝐴 ∧ Fun 𝑔))
1413simprd 113 . . . 4 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → Fun 𝑔)
15 sbthlem.1 . . . . 5 𝐴 ∈ V
16 sbthlem.2 . . . . 5 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
17 sbthlem.3 . . . . 5 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
1815, 16, 17sbthlem7 6910 . . . 4 ((Fun 𝑓 ∧ Fun 𝑔) → Fun 𝐻)
1910, 14, 18syl2anc 409 . . 3 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → Fun 𝐻)
20 simp1 982 . . . 4 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → EXMID)
219simprd 113 . . . 4 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → dom 𝑓 = 𝐴)
2213simpld 111 . . . . . 6 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝑔:𝐵𝐴)
23 df-f 5177 . . . . . 6 (𝑔:𝐵𝐴 ↔ (𝑔 Fn 𝐵 ∧ ran 𝑔𝐴))
2422, 23sylib 121 . . . . 5 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → (𝑔 Fn 𝐵 ∧ ran 𝑔𝐴))
2524simprd 113 . . . 4 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → ran 𝑔𝐴)
2615, 16, 17sbthlemi5 6908 . . . 4 ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴)) → dom 𝐻 = 𝐴)
2720, 21, 25, 26syl12anc 1218 . . 3 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → dom 𝐻 = 𝐴)
28 df-fn 5176 . . 3 (𝐻 Fn 𝐴 ↔ (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
2919, 27, 28sylanbrc 414 . 2 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝐻 Fn 𝐴)
303simprd 113 . . . 4 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → Fun 𝑓)
3124simpld 111 . . . . . 6 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝑔 Fn 𝐵)
32 df-fn 5176 . . . . . 6 (𝑔 Fn 𝐵 ↔ (Fun 𝑔 ∧ dom 𝑔 = 𝐵))
3331, 32sylib 121 . . . . 5 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → (Fun 𝑔 ∧ dom 𝑔 = 𝐵))
3433, 25jca 304 . . . 4 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴))
3515, 16, 17sbthlemi8 6911 . . . 4 (((EXMID ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → Fun 𝐻)
3620, 30, 34, 14, 35syl22anc 1221 . . 3 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → Fun 𝐻)
376simprd 113 . . . 4 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → ran 𝑓𝐵)
3833simprd 113 . . . . 5 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → dom 𝑔 = 𝐵)
3938, 25jca 304 . . . 4 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴))
40 df-rn 4600 . . . . 5 ran 𝐻 = dom 𝐻
4115, 16, 17sbthlemi6 6909 . . . . 5 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ran 𝐻 = 𝐵)
4240, 41eqtr3id 2204 . . . 4 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → dom 𝐻 = 𝐵)
4320, 37, 39, 14, 42syl22anc 1221 . . 3 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → dom 𝐻 = 𝐵)
44 df-fn 5176 . . 3 (𝐻 Fn 𝐵 ↔ (Fun 𝐻 ∧ dom 𝐻 = 𝐵))
4536, 43, 44sylanbrc 414 . 2 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝐻 Fn 𝐵)
46 dff1o4 5425 . 2 (𝐻:𝐴1-1-onto𝐵 ↔ (𝐻 Fn 𝐴𝐻 Fn 𝐵))
4729, 45, 46sylanbrc 414 1 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝐻:𝐴1-1-onto𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 963   = wceq 1335  wcel 2128  {cab 2143  Vcvv 2712  cdif 3099  cun 3100  wss 3102   cuni 3774  EXMIDwem 4158  ccnv 4588  dom cdm 4589  ran crn 4590  cres 4591  cima 4592  Fun wfun 5167   Fn wfn 5168  wf 5169  1-1wf1 5170  1-1-ontowf1o 5172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4085  ax-nul 4093  ax-pow 4138  ax-pr 4172
This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4029  df-exmid 4159  df-id 4256  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-fun 5175  df-fn 5176  df-f 5177  df-f1 5178  df-fo 5179  df-f1o 5180
This theorem is referenced by:  sbthlemi10  6913
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