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Theorem sbthlemi4 6905
Description: Lemma for isbth 6912. (Contributed by NM, 27-Mar-1998.)
Hypotheses
Ref Expression
sbthlem.1 𝐴 ∈ V
sbthlem.2 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
Assertion
Ref Expression
sbthlemi4 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝑔 “ (𝐴 𝐷)) = (𝐵 ∖ (𝑓 𝐷)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝑓   𝑥,𝑔
Allowed substitution hints:   𝐴(𝑓,𝑔)   𝐵(𝑓,𝑔)   𝐷(𝑓,𝑔)

Proof of Theorem sbthlemi4
StepHypRef Expression
1 df-ima 4600 . 2 (𝑔 “ (𝐴 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷))
2 dfdm4 4779 . . . . . 6 dom (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = ran (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷)))
3 difss 3233 . . . . . . . 8 (𝐵 ∖ (𝑓 𝐷)) ⊆ 𝐵
4 sseq2 3152 . . . . . . . 8 (dom 𝑔 = 𝐵 → ((𝐵 ∖ (𝑓 𝐷)) ⊆ dom 𝑔 ↔ (𝐵 ∖ (𝑓 𝐷)) ⊆ 𝐵))
53, 4mpbiri 167 . . . . . . 7 (dom 𝑔 = 𝐵 → (𝐵 ∖ (𝑓 𝐷)) ⊆ dom 𝑔)
6 ssdmres 4889 . . . . . . 7 ((𝐵 ∖ (𝑓 𝐷)) ⊆ dom 𝑔 ↔ dom (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = (𝐵 ∖ (𝑓 𝐷)))
75, 6sylib 121 . . . . . 6 (dom 𝑔 = 𝐵 → dom (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = (𝐵 ∖ (𝑓 𝐷)))
82, 7syl5reqr 2205 . . . . 5 (dom 𝑔 = 𝐵 → (𝐵 ∖ (𝑓 𝐷)) = ran (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))))
98adantr 274 . . . 4 ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) → (𝐵 ∖ (𝑓 𝐷)) = ran (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))))
1093ad2ant2 1004 . . 3 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝐵 ∖ (𝑓 𝐷)) = ran (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))))
11 funcnvres 5244 . . . . . . 7 (Fun 𝑔(𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = (𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))
12113ad2ant3 1005 . . . . . 6 ((EXMID ∧ ran 𝑔𝐴 ∧ Fun 𝑔) → (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = (𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))
13 sbthlem.1 . . . . . . . . 9 𝐴 ∈ V
14 sbthlem.2 . . . . . . . . 9 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
1513, 14sbthlemi3 6904 . . . . . . . 8 ((EXMID ∧ ran 𝑔𝐴) → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) = (𝐴 𝐷))
1615reseq2d 4867 . . . . . . 7 ((EXMID ∧ ran 𝑔𝐴) → (𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) = (𝑔 ↾ (𝐴 𝐷)))
17163adant3 1002 . . . . . 6 ((EXMID ∧ ran 𝑔𝐴 ∧ Fun 𝑔) → (𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) = (𝑔 ↾ (𝐴 𝐷)))
1812, 17eqtrd 2190 . . . . 5 ((EXMID ∧ ran 𝑔𝐴 ∧ Fun 𝑔) → (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = (𝑔 ↾ (𝐴 𝐷)))
1918rneqd 4816 . . . 4 ((EXMID ∧ ran 𝑔𝐴 ∧ Fun 𝑔) → ran (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = ran (𝑔 ↾ (𝐴 𝐷)))
20193adant2l 1214 . . 3 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → ran (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = ran (𝑔 ↾ (𝐴 𝐷)))
2110, 20eqtrd 2190 . 2 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝐵 ∖ (𝑓 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷)))
221, 21eqtr4id 2209 1 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝑔 “ (𝐴 𝐷)) = (𝐵 ∖ (𝑓 𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 963   = wceq 1335  wcel 2128  {cab 2143  Vcvv 2712  cdif 3099  wss 3102   cuni 3773  EXMIDwem 4156  ccnv 4586  dom cdm 4587  ran crn 4588  cres 4589  cima 4590  Fun wfun 5165
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 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170
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 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-exmid 4157  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-fun 5173
This theorem is referenced by:  sbthlemi6  6907  sbthlemi8  6909
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