Theorem sbthlemi5 7278

Description: Lemma for isbth 7284. (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 4982	. . . 4 ⊢ dom 𝐻 = dom ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
3		dmun 4988	. . . 4 ⊢ dom ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = (dom (𝑓 ↾ ∪ 𝐷) ∪ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
4		dmres 5084	. . . . 5 ⊢ dom (𝑓 ↾ ∪ 𝐷) = (∪ 𝐷 ∩ dom 𝑓)
5		dmres 5084	. . . . . 6 ⊢ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) = ((𝐴 ∖ ∪ 𝐷) ∩ dom ◡𝑔)
6		df-rn 4785	. . . . . . . 8 ⊢ ran 𝑔 = dom ◡𝑔
7	6	eqcomi 2242	. . . . . . 7 ⊢ dom ◡𝑔 = ran 𝑔
8	7	ineq2i 3429	. . . . . 6 ⊢ ((𝐴 ∖ ∪ 𝐷) ∩ dom ◡𝑔) = ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔)
9	5, 8	eqtri 2259	. . . . 5 ⊢ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) = ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔)
10	4, 9	uneq12i 3381	. . . 4 ⊢ (dom (𝑓 ↾ ∪ 𝐷) ∪ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ((∪ 𝐷 ∩ dom 𝑓) ∪ ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔))
11	2, 3, 10	3eqtri 2263	. . 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 7274	. . . . . . . . 9 ⊢ ∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
15		difss 3355	. . . . . . . . 9 ⊢ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ 𝐴
16	14, 15	sstri 3257	. . . . . . . 8 ⊢ ∪ 𝐷 ⊆ 𝐴
17		sseq2 3272	. . . . . . . 8 ⊢ (dom 𝑓 = 𝐴 → (∪ 𝐷 ⊆ dom 𝑓 ↔ ∪ 𝐷 ⊆ 𝐴))
18	16, 17	mpbiri 168	. . . . . . 7 ⊢ (dom 𝑓 = 𝐴 → ∪ 𝐷 ⊆ dom 𝑓)
19		dfss 3234	. . . . . . 7 ⊢ (∪ 𝐷 ⊆ dom 𝑓 ↔ ∪ 𝐷 = (∪ 𝐷 ∩ dom 𝑓))
20	18, 19	sylib 122	. . . . . 6 ⊢ (dom 𝑓 = 𝐴 → ∪ 𝐷 = (∪ 𝐷 ∩ dom 𝑓))
21	20	uneq1d 3382	. . . . 5 ⊢ (dom 𝑓 = 𝐴 → (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)) = ((∪ 𝐷 ∩ dom 𝑓) ∪ (𝐴 ∖ ∪ 𝐷)))
22	12, 13	sbthlemi3 7276	. . . . . . . 8 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴) → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐴 ∖ ∪ 𝐷))
23		imassrn 5137	. . . . . . . 8 ⊢ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ ran 𝑔
24	22, 23	eqsstrrdi 3301	. . . . . . 7 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴) → (𝐴 ∖ ∪ 𝐷) ⊆ ran 𝑔)
25		dfss 3234	. . . . . . 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 3383	. . . . 5 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴) → ((∪ 𝐷 ∩ dom 𝑓) ∪ (𝐴 ∖ ∪ 𝐷)) = ((∪ 𝐷 ∩ dom 𝑓) ∪ ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔)))
28	21, 27	sylan9eq 2291	. . . 4 ⊢ ((dom 𝑓 = 𝐴 ∧ (EXMID ∧ ran 𝑔 ⊆ 𝐴)) → (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)) = ((∪ 𝐷 ∩ dom 𝑓) ∪ ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔)))
29	28	an12s 571	. . 3 ⊢ ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴)) → (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)) = ((∪ 𝐷 ∩ dom 𝑓) ∪ ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔)))
30	11, 29	eqtr4id 2290	. 2 ⊢ ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴)) → dom 𝐻 = (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)))
31		undifdcss 7230	. . . . 5 ⊢ (𝐴 = (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)) ↔ (∪ 𝐷 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ ∪ 𝐷))
32		exmidexmid 4333	. . . . . . 7 ⊢ (EXMID → DECID 𝑦 ∈ ∪ 𝐷)
33	32	ralrimivw 2624	. . . . . 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 2274	1 ⊢ ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴)) → dom 𝐻 = 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104  DECID wdc 846   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  Vcvv 2821   ∖ cdif 3217   ∪ cun 3218   ∩ cin 3219   ⊆ wss 3220  ∪ cuni 3935  EXMIDwem 4331  ^◡ccnv 4773  dom cdm 4774  ran crn 4775   ↾ cres 4776   “ cima 4777

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-exmid 4332  df-xp 4780  df-cnv 4782  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787

This theorem is used by:  sbthlemi9  7282