Theorem sbthlemi5 7203

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))}
sbthlem.3	⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))

Assertion

Ref	Expression
sbthlemi5	⊢ ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴)) → dom 𝐻 = 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝑓   𝑥,𝑔   𝑥,𝐻

Allowed substitution hints:   𝐴(𝑓,𝑔)   𝐵(𝑓,𝑔)   𝐷(𝑓,𝑔)   𝐻(𝑓,𝑔)

Proof of Theorem sbthlemi5

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbthlem.3	. . . . 5 ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
2	1	dmeqi 4938	. . . 4 ⊢ dom 𝐻 = dom ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
3		dmun 4944	. . . 4 ⊢ dom ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = (dom (𝑓 ↾ ∪ 𝐷) ∪ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
4		dmres 5040	. . . . 5 ⊢ dom (𝑓 ↾ ∪ 𝐷) = (∪ 𝐷 ∩ dom 𝑓)
5		dmres 5040	. . . . . 6 ⊢ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) = ((𝐴 ∖ ∪ 𝐷) ∩ dom ◡𝑔)
6		df-rn 4742	. . . . . . . 8 ⊢ ran 𝑔 = dom ◡𝑔
7	6	eqcomi 2235	. . . . . . 7 ⊢ dom ◡𝑔 = ran 𝑔
8	7	ineq2i 3407	. . . . . 6 ⊢ ((𝐴 ∖ ∪ 𝐷) ∩ dom ◡𝑔) = ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔)
9	5, 8	eqtri 2252	. . . . 5 ⊢ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) = ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔)
10	4, 9	uneq12i 3361	. . . 4 ⊢ (dom (𝑓 ↾ ∪ 𝐷) ∪ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ((∪ 𝐷 ∩ dom 𝑓) ∪ ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔))
11	2, 3, 10	3eqtri 2256	. . 3 ⊢ dom 𝐻 = ((∪ 𝐷 ∩ dom 𝑓) ∪ ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔))
12		sbthlem.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ V
13		sbthlem.2	. . . . . . . . . 10 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}
14	12, 13	sbthlem1 7199	. . . . . . . . 9 ⊢ ∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
15		difss 3335	. . . . . . . . 9 ⊢ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ 𝐴
16	14, 15	sstri 3237	. . . . . . . 8 ⊢ ∪ 𝐷 ⊆ 𝐴
17		sseq2 3252	. . . . . . . 8 ⊢ (dom 𝑓 = 𝐴 → (∪ 𝐷 ⊆ dom 𝑓 ↔ ∪ 𝐷 ⊆ 𝐴))
18	16, 17	mpbiri 168	. . . . . . 7 ⊢ (dom 𝑓 = 𝐴 → ∪ 𝐷 ⊆ dom 𝑓)
19		dfss 3215	. . . . . . 7 ⊢ (∪ 𝐷 ⊆ dom 𝑓 ↔ ∪ 𝐷 = (∪ 𝐷 ∩ dom 𝑓))
20	18, 19	sylib 122	. . . . . 6 ⊢ (dom 𝑓 = 𝐴 → ∪ 𝐷 = (∪ 𝐷 ∩ dom 𝑓))
21	20	uneq1d 3362	. . . . 5 ⊢ (dom 𝑓 = 𝐴 → (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)) = ((∪ 𝐷 ∩ dom 𝑓) ∪ (𝐴 ∖ ∪ 𝐷)))
22	12, 13	sbthlemi3 7201	. . . . . . . 8 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴) → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐴 ∖ ∪ 𝐷))
23		imassrn 5093	. . . . . . . 8 ⊢ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ ran 𝑔
24	22, 23	eqsstrrdi 3281	. . . . . . 7 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴) → (𝐴 ∖ ∪ 𝐷) ⊆ ran 𝑔)
25		dfss 3215	. . . . . . 7 ⊢ ((𝐴 ∖ ∪ 𝐷) ⊆ ran 𝑔 ↔ (𝐴 ∖ ∪ 𝐷) = ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔))
26	24, 25	sylib 122	. . . . . 6 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴) → (𝐴 ∖ ∪ 𝐷) = ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔))
27	26	uneq2d 3363	. . . . 5 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴) → ((∪ 𝐷 ∩ dom 𝑓) ∪ (𝐴 ∖ ∪ 𝐷)) = ((∪ 𝐷 ∩ dom 𝑓) ∪ ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔)))
28	21, 27	sylan9eq 2284	. . . 4 ⊢ ((dom 𝑓 = 𝐴 ∧ (EXMID ∧ ran 𝑔 ⊆ 𝐴)) → (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)) = ((∪ 𝐷 ∩ dom 𝑓) ∪ ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔)))
29	28	an12s 567	. . 3 ⊢ ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴)) → (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)) = ((∪ 𝐷 ∩ dom 𝑓) ∪ ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔)))
30	11, 29	eqtr4id 2283	. 2 ⊢ ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴)) → dom 𝐻 = (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)))
31		undifdcss 7158	. . . . 5 ⊢ (𝐴 = (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)) ↔ (∪ 𝐷 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ ∪ 𝐷))
32		exmidexmid 4292	. . . . . . 7 ⊢ (EXMID → DECID 𝑦 ∈ ∪ 𝐷)
33	32	ralrimivw 2607	. . . . . 6 ⊢ (EXMID → ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ ∪ 𝐷)
34	33	biantrud 304	. . . . 5 ⊢ (EXMID → (∪ 𝐷 ⊆ 𝐴 ↔ (∪ 𝐷 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ ∪ 𝐷)))
35	31, 34	bitr4id 199	. . . 4 ⊢ (EXMID → (𝐴 = (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)) ↔ ∪ 𝐷 ⊆ 𝐴))
36	16, 35	mpbiri 168	. . 3 ⊢ (EXMID → 𝐴 = (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)))
37	36	adantr 276	. 2 ⊢ ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴)) → 𝐴 = (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)))
38	30, 37	eqtr4d 2267	1 ⊢ ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴)) → dom 𝐻 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 842   = wceq 1398   ∈ wcel 2202  {cab 2217  ∀wral 2511  Vcvv 2803   ∖ cdif 3198   ∪ cun 3199   ∩ cin 3200   ⊆ wss 3201  ∪ cuni 3898  EXMIDwem 4290  ^◡ccnv 4730  dom cdm 4731  ran crn 4732   ↾ cres 4733   “ 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:  sbthlemi9  7207