Theorem sbthlemi3 7201

Description: Lemma for isbth 7209. (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
sbthlemi3	⊢ ((EXMID ∧ 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 sbthlemi3

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbthlem.1	. . . . . . 7 ⊢ 𝐴 ∈ V
2		sbthlem.2	. . . . . . 7 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}
3	1, 2	sbthlem2 7200	. . . . . 6 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ ∪ 𝐷)
4	1, 2	sbthlem1 7199	. . . . . 6 ⊢ ∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
5	3, 4	jctil 312	. . . . 5 ⊢ (ran 𝑔 ⊆ 𝐴 → (∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ∧ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ ∪ 𝐷))
6		eqss 3243	. . . . 5 ⊢ (∪ 𝐷 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ↔ (∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ∧ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ ∪ 𝐷))
7	5, 6	sylibr 134	. . . 4 ⊢ (ran 𝑔 ⊆ 𝐴 → ∪ 𝐷 = (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
8	7	difeq2d 3327	. . 3 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝐴 ∖ ∪ 𝐷) = (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))
9	8	adantl 277	. 2 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴) → (𝐴 ∖ ∪ 𝐷) = (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))))
10		imassrn 5093	. . . . 5 ⊢ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ ran 𝑔
11		sstr2 3235	. . . . 5 ⊢ ((𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ ran 𝑔 → (ran 𝑔 ⊆ 𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ 𝐴))
12	10, 11	ax-mp 5	. . . 4 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ 𝐴)
13		exmidexmid 4292	. . . . . . 7 ⊢ (EXMID → DECID 𝑦 ∈ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
14		dcstab 852	. . . . . . 7 ⊢ (DECID 𝑦 ∈ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) → STAB 𝑦 ∈ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
15	13, 14	syl 14	. . . . . 6 ⊢ (EXMID → STAB 𝑦 ∈ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
16	15	alrimiv 1922	. . . . 5 ⊢ (EXMID → ∀𝑦STAB 𝑦 ∈ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
17		dfss4st 3442	. . . . 5 ⊢ (∀𝑦STAB 𝑦 ∈ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) → ((𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))) = (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
18	16, 17	syl 14	. . . 4 ⊢ (EXMID → ((𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))) = (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
19	12, 18	imbitrid 154	. . 3 ⊢ (EXMID → (ran 𝑔 ⊆ 𝐴 → (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))) = (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
20	19	imp 124	. 2 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴) → (𝐴 ∖ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))) = (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
21	9, 20	eqtr2d 2265	1 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴) → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐴 ∖ ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  STAB wstab 838  DECID wdc 842  ∀wal 1396   = wceq 1398   ∈ wcel 2202  {cab 2217  Vcvv 2803   ∖ cdif 3198   ⊆ wss 3201  ∪ cuni 3898  EXMIDwem 4290  ran crn 4732   “ cima 4734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-exmid 4291  df-xp 4737  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744

This theorem is referenced by:  sbthlemi4  7202  sbthlemi5  7203