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Theorem sbthlem3 8188
Description: Lemma for sbth 8196. (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 8187 . . . . 5 (ran 𝑔𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐷)
41, 2sbthlem1 8186 . . . . 5 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
53, 4jctil 561 . . . 4 (ran 𝑔𝐴 → ( 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ∧ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐷))
6 eqss 3724 . . . 4 ( 𝐷 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ↔ ( 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ∧ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐷))
75, 6sylibr 224 . . 3 (ran 𝑔𝐴 𝐷 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))
87difeq2d 3836 . 2 (ran 𝑔𝐴 → (𝐴 𝐷) = (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))))
9 imassrn 5587 . . . 4 (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ ran 𝑔
10 sstr2 3716 . . . 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 3966 . . 3 ((𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))) = (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
1311, 12sylib 208 . 2 (ran 𝑔𝐴 → (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))) = (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
148, 13eqtr2d 2759 1 (ran 𝑔𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) = (𝐴 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wcel 2103  {cab 2710  Vcvv 3304  cdif 3677  wss 3680   cuni 4544  ran crn 5219  cima 5221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-xp 5224  df-cnv 5226  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231
This theorem is referenced by:  sbthlem4  8189  sbthlem5  8190
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