Theorem sbthlem3 9011

Description: Lemma for sbth 9019. (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 9010	. . . . 5 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ ∪ 𝐷)
4	1, 2	sbthlem1 9009	. . . . 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 3946	. . . 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 4075	. 2 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝐴 ∖ ∪ 𝐷) = (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))
9		imassrn 6026	. . . 4 ⊢ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ ran 𝑔
10		sstr2 3937	. . . 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 4218	. . 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 2769	1 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐴 ∖ ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2711  Vcvv 3437   ∖ cdif 3895   ⊆ wss 3898  ∪ cuni 4860  ran crn 5622   “ cima 5624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-xp 5627  df-cnv 5629  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634

This theorem is referenced by:  sbthlem4  9012  sbthlem5  9013