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Theorem sbthlem4 8114
 Description: Lemma for sbth 8121. (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 dfdm4 5348 . . . . 5 dom (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = ran (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷)))
2 difss 3770 . . . . . . 7 (𝐵 ∖ (𝑓 𝐷)) ⊆ 𝐵
3 sseq2 3660 . . . . . . 7 (dom 𝑔 = 𝐵 → ((𝐵 ∖ (𝑓 𝐷)) ⊆ dom 𝑔 ↔ (𝐵 ∖ (𝑓 𝐷)) ⊆ 𝐵))
42, 3mpbiri 248 . . . . . 6 (dom 𝑔 = 𝐵 → (𝐵 ∖ (𝑓 𝐷)) ⊆ dom 𝑔)
5 ssdmres 5455 . . . . . 6 ((𝐵 ∖ (𝑓 𝐷)) ⊆ dom 𝑔 ↔ dom (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = (𝐵 ∖ (𝑓 𝐷)))
64, 5sylib 208 . . . . 5 (dom 𝑔 = 𝐵 → dom (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = (𝐵 ∖ (𝑓 𝐷)))
71, 6syl5reqr 2700 . . . 4 (dom 𝑔 = 𝐵 → (𝐵 ∖ (𝑓 𝐷)) = ran (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))))
8 funcnvres 6005 . . . . . 6 (Fun 𝑔(𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = (𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))))
9 sbthlem.1 . . . . . . . 8 𝐴 ∈ V
10 sbthlem.2 . . . . . . . 8 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
119, 10sbthlem3 8113 . . . . . . 7 (ran 𝑔𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) = (𝐴 𝐷))
1211reseq2d 5428 . . . . . 6 (ran 𝑔𝐴 → (𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) = (𝑔 ↾ (𝐴 𝐷)))
138, 12sylan9eqr 2707 . . . . 5 ((ran 𝑔𝐴 ∧ Fun 𝑔) → (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = (𝑔 ↾ (𝐴 𝐷)))
1413rneqd 5385 . . . 4 ((ran 𝑔𝐴 ∧ Fun 𝑔) → ran (𝑔 ↾ (𝐵 ∖ (𝑓 𝐷))) = ran (𝑔 ↾ (𝐴 𝐷)))
157, 14sylan9eq 2705 . . 3 ((dom 𝑔 = 𝐵 ∧ (ran 𝑔𝐴 ∧ Fun 𝑔)) → (𝐵 ∖ (𝑓 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷)))
1615anassrs 681 . 2 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝐵 ∖ (𝑓 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷)))
17 df-ima 5156 . 2 (𝑔 “ (𝐴 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷))
1816, 17syl6reqr 2704 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 383   = wceq 1523   ∈ wcel 2030  {cab 2637  Vcvv 3231   ∖ cdif 3604   ⊆ wss 3607  ∪ cuni 4468  ◡ccnv 5142  dom cdm 5143  ran crn 5144   ↾ cres 5145   “ cima 5146  Fun wfun 5920 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-fun 5928 This theorem is referenced by:  sbthlem6  8116  sbthlem8  8118
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