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Theorem sbthlemi9 6846
Description: Lemma for isbth 6848. (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 982 . . . . . . . . . 10 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝑓:𝐴1-1𝐵)
2 df-f1 5123 . . . . . . . . . 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 5122 . . . . . . . 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 5121 . . . . . 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 983 . . . . . 6 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝑔:𝐵1-1𝐴)
12 df-f1 5123 . . . . . 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 6844 . . . 4 ((Fun 𝑓 ∧ Fun 𝑔) → Fun 𝐻)
1910, 14, 18syl2anc 408 . . 3 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → Fun 𝐻)
20 simp1 981 . . . 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 5122 . . . . . 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 6842 . . . 4 ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴)) → dom 𝐻 = 𝐴)
2720, 21, 25, 26syl12anc 1214 . . 3 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → dom 𝐻 = 𝐴)
28 df-fn 5121 . . 3 (𝐻 Fn 𝐴 ↔ (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
2919, 27, 28sylanbrc 413 . 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 5121 . . . . . 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 6845 . . . 4 (((EXMID ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → Fun 𝐻)
3620, 30, 34, 14, 35syl22anc 1217 . . 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 4545 . . . . 5 ran 𝐻 = dom 𝐻
4115, 16, 17sbthlemi6 6843 . . . . 5 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ran 𝐻 = 𝐵)
4240, 41syl5eqr 2184 . . . 4 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → dom 𝐻 = 𝐵)
4320, 37, 39, 14, 42syl22anc 1217 . . 3 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → dom 𝐻 = 𝐵)
44 df-fn 5121 . . 3 (𝐻 Fn 𝐵 ↔ (Fun 𝐻 ∧ dom 𝐻 = 𝐵))
4536, 43, 44sylanbrc 413 . 2 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝐻 Fn 𝐵)
46 dff1o4 5368 . 2 (𝐻:𝐴1-1-onto𝐵 ↔ (𝐻 Fn 𝐴𝐻 Fn 𝐵))
4729, 45, 46sylanbrc 413 1 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝐻:𝐴1-1-onto𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 962   = wceq 1331  wcel 1480  {cab 2123  Vcvv 2681  cdif 3063  cun 3064  wss 3066   cuni 3731  EXMIDwem 4113  ccnv 4533  dom cdm 4534  ran crn 4535  cres 4536  cima 4537  Fun wfun 5112   Fn wfn 5113  wf 5114  1-1wf1 5115  1-1-ontowf1o 5117
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-exmid 4114  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125
This theorem is referenced by:  sbthlemi10  6847
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