Theorem sbthlem3 9108

Description: Lemma for sbth 9116. (Contributed by NM, 22-Mar-1998.)

Hypotheses

Ref	Expression
sbthlem.1	⊢ 𝐴 ∈ V
sbthlem.2	⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}

Assertion

Ref	Expression
sbthlem3	⊢ (ran 𝑔 ⊆ 𝐴 → (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

Step	Hyp	Ref	Expression
1		sbthlem.1	. . . . . 6 ⊢ 𝐴 ∈ V
2		sbthlem.2	. . . . . 6 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}
3	1, 2	sbthlem2 9107	. . . . 5 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ ∪ 𝐷)
4	1, 2	sbthlem1 9106	. . . . 5 ⊢ ∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
5	3, 4	jctil 519	. . . 4 ⊢ (ran 𝑔 ⊆ 𝐴 → (∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ∧ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ ∪ 𝐷))
6		eqss 3981	. . . 4 ⊢ (∪ 𝐷 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ↔ (∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ∧ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ ∪ 𝐷))
7	5, 6	sylibr 234	. . 3 ⊢ (ran 𝑔 ⊆ 𝐴 → ∪ 𝐷 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
8	7	difeq2d 4108	. 2 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝐴 ∖ ∪ 𝐷) = (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))
9		imassrn 6071	. . . 4 ⊢ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ ran 𝑔
10		sstr2 3972	. . . 4 ⊢ ((𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ ran 𝑔 → (ran 𝑔 ⊆ 𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ 𝐴))
11	9, 10	ax-mp 5	. . 3 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ 𝐴)
12		dfss4 4251	. . 3 ⊢ ((𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))) = (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
13	11, 12	sylib 218	. 2 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))) = (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
14	8, 13	eqtr2d 2770	1 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐴 ∖ ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {cab 2712  Vcvv 3464   ∖ cdif 3930   ⊆ wss 3933  ∪ cuni 4889  ran crn 5668   “ cima 5670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-br 5126  df-opab 5188  df-xp 5673  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680

This theorem is referenced by:  sbthlem4  9109  sbthlem5  9110