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Theorem sbthlemi4 6814
 Description: Lemma for isbth 6821. (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 dfdm4 4699 . . . . . 6 dom (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = ran (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷)))
2 difss 3170 . . . . . . . 8 (𝐵 ∖ (𝑓 𝐷)) ⊆ 𝐵
3 sseq2 3089 . . . . . . . 8 (dom 𝑔 = 𝐵 → ((𝐵 ∖ (𝑓 𝐷)) ⊆ dom 𝑔 ↔ (𝐵 ∖ (𝑓 𝐷)) ⊆ 𝐵))
42, 3mpbiri 167 . . . . . . 7 (dom 𝑔 = 𝐵 → (𝐵 ∖ (𝑓 𝐷)) ⊆ dom 𝑔)
5 ssdmres 4809 . . . . . . 7 ((𝐵 ∖ (𝑓 𝐷)) ⊆ dom 𝑔 ↔ dom (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = (𝐵 ∖ (𝑓 𝐷)))
64, 5sylib 121 . . . . . 6 (dom 𝑔 = 𝐵 → dom (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = (𝐵 ∖ (𝑓 𝐷)))
71, 6syl5reqr 2163 . . . . 5 (dom 𝑔 = 𝐵 → (𝐵 ∖ (𝑓 𝐷)) = ran (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))))
87adantr 272 . . . 4 ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) → (𝐵 ∖ (𝑓 𝐷)) = ran (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))))
983ad2ant2 986 . . 3 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝐵 ∖ (𝑓 𝐷)) = ran (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))))
10 funcnvres 5164 . . . . . . 7 (Fun 𝑔(𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = (𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))
11103ad2ant3 987 . . . . . 6 ((EXMID ∧ ran 𝑔𝐴 ∧ Fun 𝑔) → (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = (𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))
12 sbthlem.1 . . . . . . . . 9 𝐴 ∈ V
13 sbthlem.2 . . . . . . . . 9 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
1412, 13sbthlemi3 6813 . . . . . . . 8 ((EXMID ∧ ran 𝑔𝐴) → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) = (𝐴 𝐷))
1514reseq2d 4787 . . . . . . 7 ((EXMID ∧ ran 𝑔𝐴) → (𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) = (𝑔 ↾ (𝐴 𝐷)))
16153adant3 984 . . . . . 6 ((EXMID ∧ ran 𝑔𝐴 ∧ Fun 𝑔) → (𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) = (𝑔 ↾ (𝐴 𝐷)))
1711, 16eqtrd 2148 . . . . 5 ((EXMID ∧ ran 𝑔𝐴 ∧ Fun 𝑔) → (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = (𝑔 ↾ (𝐴 𝐷)))
1817rneqd 4736 . . . 4 ((EXMID ∧ ran 𝑔𝐴 ∧ Fun 𝑔) → ran (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = ran (𝑔 ↾ (𝐴 𝐷)))
19183adant2l 1193 . . 3 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → ran (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = ran (𝑔 ↾ (𝐴 𝐷)))
209, 19eqtrd 2148 . 2 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝐵 ∖ (𝑓 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷)))
21 df-ima 4520 . 2 (𝑔 “ (𝐴 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷))
2220, 21syl6reqr 2167 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 945   = wceq 1314   ∈ wcel 1463  {cab 2101  Vcvv 2658   ∖ cdif 3036   ⊆ wss 3039  ∪ cuni 3704  EXMIDwem 4086  ◡ccnv 4506  dom cdm 4507  ran crn 4508   ↾ cres 4509   “ cima 4510  Fun wfun 5085 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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099 This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-exmid 4087  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-fun 5093 This theorem is referenced by:  sbthlemi6  6816  sbthlemi8  6818
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