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Theorem sbthlemi9 6977
Description: Lemma for isbth 6979. (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 999 . . . . . . . . . 10 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝑓:𝐴1-1𝐵)
2 df-f1 5233 . . . . . . . . . 10 (𝑓:𝐴1-1𝐵 ↔ (𝑓:𝐴𝐵 ∧ Fun 𝑓))
31, 2sylib 122 . . . . . . . . 9 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → (𝑓:𝐴𝐵 ∧ Fun 𝑓))
43simpld 112 . . . . . . . 8 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝑓:𝐴𝐵)
5 df-f 5232 . . . . . . . 8 (𝑓:𝐴𝐵 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓𝐵))
64, 5sylib 122 . . . . . . 7 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → (𝑓 Fn 𝐴 ∧ ran 𝑓𝐵))
76simpld 112 . . . . . 6 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝑓 Fn 𝐴)
8 df-fn 5231 . . . . . 6 (𝑓 Fn 𝐴 ↔ (Fun 𝑓 ∧ dom 𝑓 = 𝐴))
97, 8sylib 122 . . . . 5 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → (Fun 𝑓 ∧ dom 𝑓 = 𝐴))
109simpld 112 . . . 4 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → Fun 𝑓)
11 simp3 1000 . . . . . 6 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝑔:𝐵1-1𝐴)
12 df-f1 5233 . . . . . 6 (𝑔:𝐵1-1𝐴 ↔ (𝑔:𝐵𝐴 ∧ Fun 𝑔))
1311, 12sylib 122 . . . . 5 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → (𝑔:𝐵𝐴 ∧ Fun 𝑔))
1413simprd 114 . . . 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 6975 . . . 4 ((Fun 𝑓 ∧ Fun 𝑔) → Fun 𝐻)
1910, 14, 18syl2anc 411 . . 3 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → Fun 𝐻)
20 simp1 998 . . . 4 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → EXMID)
219simprd 114 . . . 4 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → dom 𝑓 = 𝐴)
2213simpld 112 . . . . . 6 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝑔:𝐵𝐴)
23 df-f 5232 . . . . . 6 (𝑔:𝐵𝐴 ↔ (𝑔 Fn 𝐵 ∧ ran 𝑔𝐴))
2422, 23sylib 122 . . . . 5 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → (𝑔 Fn 𝐵 ∧ ran 𝑔𝐴))
2524simprd 114 . . . 4 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → ran 𝑔𝐴)
2615, 16, 17sbthlemi5 6973 . . . 4 ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴)) → dom 𝐻 = 𝐴)
2720, 21, 25, 26syl12anc 1246 . . 3 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → dom 𝐻 = 𝐴)
28 df-fn 5231 . . 3 (𝐻 Fn 𝐴 ↔ (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
2919, 27, 28sylanbrc 417 . 2 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝐻 Fn 𝐴)
303simprd 114 . . . 4 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → Fun 𝑓)
3124simpld 112 . . . . . 6 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝑔 Fn 𝐵)
32 df-fn 5231 . . . . . 6 (𝑔 Fn 𝐵 ↔ (Fun 𝑔 ∧ dom 𝑔 = 𝐵))
3331, 32sylib 122 . . . . 5 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → (Fun 𝑔 ∧ dom 𝑔 = 𝐵))
3433, 25jca 306 . . . 4 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴))
3515, 16, 17sbthlemi8 6976 . . . 4 (((EXMID ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → Fun 𝐻)
3620, 30, 34, 14, 35syl22anc 1249 . . 3 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → Fun 𝐻)
376simprd 114 . . . 4 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → ran 𝑓𝐵)
3833simprd 114 . . . . 5 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → dom 𝑔 = 𝐵)
3938, 25jca 306 . . . 4 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴))
40 df-rn 4649 . . . . 5 ran 𝐻 = dom 𝐻
4115, 16, 17sbthlemi6 6974 . . . . 5 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ran 𝐻 = 𝐵)
4240, 41eqtr3id 2234 . . . 4 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → dom 𝐻 = 𝐵)
4320, 37, 39, 14, 42syl22anc 1249 . . 3 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → dom 𝐻 = 𝐵)
44 df-fn 5231 . . 3 (𝐻 Fn 𝐵 ↔ (Fun 𝐻 ∧ dom 𝐻 = 𝐵))
4536, 43, 44sylanbrc 417 . 2 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝐻 Fn 𝐵)
46 dff1o4 5481 . 2 (𝐻:𝐴1-1-onto𝐵 ↔ (𝐻 Fn 𝐴𝐻 Fn 𝐵))
4729, 45, 46sylanbrc 417 1 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝐻:𝐴1-1-onto𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 979   = wceq 1363  wcel 2158  {cab 2173  Vcvv 2749  cdif 3138  cun 3139  wss 3141   cuni 3821  EXMIDwem 4206  ccnv 4637  dom cdm 4638  ran crn 4639  cres 4640  cima 4641  Fun wfun 5222   Fn wfn 5223  wf 5224  1-1wf1 5225  1-1-ontowf1o 5227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-exmid 4207  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235
This theorem is referenced by:  sbthlemi10  6978
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