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Theorem sbthlem4 9088
Description: Lemma for sbth 9095. (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
sbthlem4 (((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 sbthlem4
StepHypRef Expression
1 df-ima 5689 . 2 (𝑔 “ (𝐴 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷))
2 difss 4131 . . . . . . 7 (𝐵 ∖ (𝑓 𝐷)) ⊆ 𝐵
3 sseq2 4008 . . . . . . 7 (dom 𝑔 = 𝐵 → ((𝐵 ∖ (𝑓 𝐷)) ⊆ dom 𝑔 ↔ (𝐵 ∖ (𝑓 𝐷)) ⊆ 𝐵))
42, 3mpbiri 257 . . . . . 6 (dom 𝑔 = 𝐵 → (𝐵 ∖ (𝑓 𝐷)) ⊆ dom 𝑔)
5 ssdmres 6004 . . . . . 6 ((𝐵 ∖ (𝑓 𝐷)) ⊆ dom 𝑔 ↔ dom (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = (𝐵 ∖ (𝑓 𝐷)))
64, 5sylib 217 . . . . 5 (dom 𝑔 = 𝐵 → dom (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = (𝐵 ∖ (𝑓 𝐷)))
7 dfdm4 5895 . . . . 5 dom (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = ran (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷)))
86, 7eqtr3di 2787 . . . 4 (dom 𝑔 = 𝐵 → (𝐵 ∖ (𝑓 𝐷)) = ran (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))))
9 funcnvres 6626 . . . . . 6 (Fun 𝑔(𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = (𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))
10 sbthlem.1 . . . . . . . 8 𝐴 ∈ V
11 sbthlem.2 . . . . . . . 8 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
1210, 11sbthlem3 9087 . . . . . . 7 (ran 𝑔𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) = (𝐴 𝐷))
1312reseq2d 5981 . . . . . 6 (ran 𝑔𝐴 → (𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) = (𝑔 ↾ (𝐴 𝐷)))
149, 13sylan9eqr 2794 . . . . 5 ((ran 𝑔𝐴 ∧ Fun 𝑔) → (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = (𝑔 ↾ (𝐴 𝐷)))
1514rneqd 5937 . . . 4 ((ran 𝑔𝐴 ∧ Fun 𝑔) → ran (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = ran (𝑔 ↾ (𝐴 𝐷)))
168, 15sylan9eq 2792 . . 3 ((dom 𝑔 = 𝐵 ∧ (ran 𝑔𝐴 ∧ Fun 𝑔)) → (𝐵 ∖ (𝑓 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷)))
1716anassrs 468 . 2 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝐵 ∖ (𝑓 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷)))
181, 17eqtr4id 2791 1 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝑔 “ (𝐴 𝐷)) = (𝐵 ∖ (𝑓 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {cab 2709  Vcvv 3474  cdif 3945  wss 3948   cuni 4908  ccnv 5675  dom cdm 5676  ran crn 5677  cres 5678  cima 5679  Fun wfun 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-fun 6545
This theorem is referenced by:  sbthlem6  9090  sbthlem8  9092
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