Theorem sbthlem2 8680

Description: Lemma for sbth 8689. (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 8679	. . . . . . . 8 ⊢ ∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
4		imass2 5939	. . . . . . . 8 ⊢ (∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) → (𝑓 “ ∪ 𝐷) ⊆ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))
5		sscon 4029	. . . . . . . 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 5939	. . . . . . 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 4029	. . . . . . 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 5914	. . . . . . . 8 ⊢ (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))) ⊆ ran 𝑔
11		sstr2 3884	. . . . . . . 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 4022	. . . . . . 7 ⊢ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ 𝐴
14		ssconb 4028	. . . . . . 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 589	. . . . . 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 523	. . . 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 5196	. . . . 5 ⊢ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ∈ V
19		sseq1 3902	. . . . . 6 ⊢ (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) → (𝑥 ⊆ 𝐴 ↔ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ 𝐴))
20		imaeq2 5899	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) → (𝑓 “ 𝑥) = (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))
21	20	difeq2d 4013	. . . . . . . 8 ⊢ (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) → (𝐵 ∖ (𝑓 “ 𝑥)) = (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))))
22	21	imaeq2d 5903	. . . . . . 7 ⊢ (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) → (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) = (𝑔 “ (𝐵 ∖ (𝑓 “ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))))
23		difeq2 4007	. . . . . . 7 ⊢ (𝑥 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) → (𝐴 ∖ 𝑥) = (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))
24	22, 23	sseq12d 3910	. . . . . 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 634	. . . . 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 3573	. . . 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 2844	. 2 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ∈ 𝐷)
29		elssuni 4828	. 2 ⊢ ((𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ∈ 𝐷 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ ∪ 𝐷)
30	28, 29	syl 17	1 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ ∪ 𝐷)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1542   ∈ wcel 2114  {cab 2716  Vcvv 3398   ∖ cdif 3840   ⊆ wss 3843  ∪ cuni 4796  ran crn 5526   “ cima 5528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-xp 5531  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538

This theorem is referenced by:  sbthlem3  8681