Theorem sbthlem2 9077

Description: Lemma for sbth 9086. (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
sbthlem2	⊢ (ran 𝑔 ⊆ 𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ ∪ 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 sbthlem2

Step	Hyp	Ref	Expression
1		sbthlem.1	. . . . . . . . 9 ⊢ 𝐴 ∈ V
2		sbthlem.2	. . . . . . . . 9 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}
3	1, 2	sbthlem1 9076	. . . . . . . 8 ⊢ ∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
4		imass2 6106	. . . . . . . 8 ⊢ (∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) → (𝑓 “ ∪ 𝐷) ⊆ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))
5		sscon 4098	. . . . . . . 8 ⊢ ((𝑓 “ ∪ 𝐷) ⊆ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))) → (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))) ⊆ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
6	3, 4, 5	mp2b 10	. . . . . . 7 ⊢ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))) ⊆ (𝐵 ∖ (𝑓 “ ∪ 𝐷))
7		imass2 6106	. . . . . . 7 ⊢ ((𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))) ⊆ (𝐵 ∖ (𝑓 “ ∪ 𝐷)) → (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))) ⊆ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
8		sscon 4098	. . . . . . 7 ⊢ ((𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))) ⊆ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))))))
9	6, 7, 8	mp2b 10	. . . . . 6 ⊢ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))))
10		imassrn 6075	. . . . . . . 8 ⊢ (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))) ⊆ ran 𝑔
11		sstr2 3945	. . . . . . . 8 ⊢ ((𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))) ⊆ ran 𝑔 → (ran 𝑔 ⊆ 𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))) ⊆ 𝐴))
12	10, 11	ax-mp 5	. . . . . . 7 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))) ⊆ 𝐴)
13		difss 4091	. . . . . . 7 ⊢ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ 𝐴
14		ssconb 4097	. . . . . . 7 ⊢ (((𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))) ⊆ 𝐴 ∧ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ 𝐴) → ((𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))) ⊆ (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))) ↔ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))))))
15	12, 13, 14	sylancl 597	. . . . . 6 ⊢ (ran 𝑔 ⊆ 𝐴 → ((𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))) ⊆ (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))) ↔ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))))))
16	9, 15	mpbiri 261	. . . . 5 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))) ⊆ (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))
17	16, 13	jctil 528	. . . 4 ⊢ (ran 𝑔 ⊆ 𝐴 → ((𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))) ⊆ (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))))
18	1	difexi 5302	. . . . 5 ⊢ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ∈ V
19		sseq1 3963	. . . . . 6 ⊢ (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) → (𝑥 ⊆ 𝐴 ↔ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ 𝐴))
20		imaeq2 6060	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) → (𝑓 “ 𝑥) = (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))
21	20	difeq2d 4082	. . . . . . . 8 ⊢ (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) → (𝐵 ∖ (𝑓 “ 𝑥)) = (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))))
22	21	imaeq2d 6064	. . . . . . 7 ⊢ (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) → (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) = (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))))
23		difeq2 4076	. . . . . . 7 ⊢ (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) → (𝐴 ∖ 𝑥) = (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))
24	22, 23	sseq12d 3971	. . . . . 6 ⊢ (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) → ((𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥) ↔ (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))) ⊆ (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))))
25	19, 24	anbi12d 643	. . . . 5 ⊢ (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) → ((𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥)) ↔ ((𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))) ⊆ (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))))
26	18, 25	elab 3639	. . . 4 ⊢ ((𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ∈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} ↔ ((𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))) ⊆ (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))))
27	17, 26	sylibr 237	. . 3 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ∈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))})
28	27, 2	eleqtrrdi 2874	. 2 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ∈ 𝐷)
29		elssuni 4905	. 2 ⊢ ((𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ∈ 𝐷 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ ∪ 𝐷)
30	28, 29	syl 18	1 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ ∪ 𝐷)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {cab 2741  Vcvv 3455   ∖ cdif 3903   ⊆ wss 3906  ∪ cuni 4873  ran crn 5664   “ cima 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676

This theorem is referenced by:  sbthlem3  9078