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Theorem sbthlemi9 6942
Description: Lemma for isbth 6944. (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 993 . . . . . . . . . 10 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝑓:𝐴1-1𝐵)
2 df-f1 5203 . . . . . . . . . 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 5202 . . . . . . . 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 5201 . . . . . 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 994 . . . . . 6 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝑔:𝐵1-1𝐴)
12 df-f1 5203 . . . . . 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 6940 . . . 4 ((Fun 𝑓 ∧ Fun 𝑔) → Fun 𝐻)
1910, 14, 18syl2anc 409 . . 3 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → Fun 𝐻)
20 simp1 992 . . . 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 5202 . . . . . 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 6938 . . . 4 ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴)) → dom 𝐻 = 𝐴)
2720, 21, 25, 26syl12anc 1231 . . 3 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → dom 𝐻 = 𝐴)
28 df-fn 5201 . . 3 (𝐻 Fn 𝐴 ↔ (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
2919, 27, 28sylanbrc 415 . 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 5201 . . . . . 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 6941 . . . 4 (((EXMID ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → Fun 𝐻)
3620, 30, 34, 14, 35syl22anc 1234 . . 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 4622 . . . . 5 ran 𝐻 = dom 𝐻
4115, 16, 17sbthlemi6 6939 . . . . 5 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ran 𝐻 = 𝐵)
4240, 41eqtr3id 2217 . . . 4 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → dom 𝐻 = 𝐵)
4320, 37, 39, 14, 42syl22anc 1234 . . 3 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → dom 𝐻 = 𝐵)
44 df-fn 5201 . . 3 (𝐻 Fn 𝐵 ↔ (Fun 𝐻 ∧ dom 𝐻 = 𝐵))
4536, 43, 44sylanbrc 415 . 2 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝐻 Fn 𝐵)
46 dff1o4 5450 . 2 (𝐻:𝐴1-1-onto𝐵 ↔ (𝐻 Fn 𝐴𝐻 Fn 𝐵))
4729, 45, 46sylanbrc 415 1 ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝐻:𝐴1-1-onto𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 973   = wceq 1348  wcel 2141  {cab 2156  Vcvv 2730  cdif 3118  cun 3119  wss 3121   cuni 3796  EXMIDwem 4180  ccnv 4610  dom cdm 4611  ran crn 4612  cres 4613  cima 4614  Fun wfun 5192   Fn wfn 5193  wf 5194  1-1wf1 5195  1-1-ontowf1o 5197
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-exmid 4181  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205
This theorem is referenced by:  sbthlemi10  6943
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