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Theorem sbthlem3 9036
Description: Lemma for sbth 9044. (Contributed by NM, 22-Mar-1998.)
Hypotheses
Ref Expression
sbthlem.1 𝐴 ∈ V
sbthlem.2 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
Assertion
Ref Expression
sbthlem3 (ran 𝑔𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) = (𝐴 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 sbthlem3
StepHypRef Expression
1 sbthlem.1 . . . . . 6 𝐴 ∈ V
2 sbthlem.2 . . . . . 6 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
31, 2sbthlem2 9035 . . . . 5 (ran 𝑔𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐷)
41, 2sbthlem1 9034 . . . . 5 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
53, 4jctil 521 . . . 4 (ran 𝑔𝐴 → ( 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ∧ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐷))
6 eqss 3964 . . . 4 ( 𝐷 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ↔ ( 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ∧ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐷))
75, 6sylibr 233 . . 3 (ran 𝑔𝐴 𝐷 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))
87difeq2d 4087 . 2 (ran 𝑔𝐴 → (𝐴 𝐷) = (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))
9 imassrn 6029 . . . 4 (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ ran 𝑔
10 sstr2 3956 . . . 4 ((𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ ran 𝑔 → (ran 𝑔𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ 𝐴))
119, 10ax-mp 5 . . 3 (ran 𝑔𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ 𝐴)
12 dfss4 4223 . . 3 ((𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))) = (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
1311, 12sylib 217 . 2 (ran 𝑔𝐴 → (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))) = (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
148, 13eqtr2d 2778 1 (ran 𝑔𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) = (𝐴 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {cab 2714  Vcvv 3448  cdif 3912  wss 3915   cuni 4870  ran crn 5639  cima 5641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651
This theorem is referenced by:  sbthlem4  9037  sbthlem5  9038
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