Theorem sbthlemi5 6942

Description: Lemma for isbth 6948. (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 4813	. . . 4 ⊢ dom 𝐻 = dom ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
3		dmun 4819	. . . 4 ⊢ dom ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = (dom (𝑓 ↾ ∪ 𝐷) ∪ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
4		dmres 4913	. . . . 5 ⊢ dom (𝑓 ↾ ∪ 𝐷) = (∪ 𝐷 ∩ dom 𝑓)
5		dmres 4913	. . . . . 6 ⊢ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) = ((𝐴 ∖ ∪ 𝐷) ∩ dom ◡𝑔)
6		df-rn 4623	. . . . . . . 8 ⊢ ran 𝑔 = dom ◡𝑔
7	6	eqcomi 2175	. . . . . . 7 ⊢ dom ◡𝑔 = ran 𝑔
8	7	ineq2i 3326	. . . . . 6 ⊢ ((𝐴 ∖ ∪ 𝐷) ∩ dom ◡𝑔) = ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔)
9	5, 8	eqtri 2192	. . . . 5 ⊢ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) = ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔)
10	4, 9	uneq12i 3280	. . . 4 ⊢ (dom (𝑓 ↾ ∪ 𝐷) ∪ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ((∪ 𝐷 ∩ dom 𝑓) ∪ ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔))
11	2, 3, 10	3eqtri 2196	. . 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 6938	. . . . . . . . 9 ⊢ ∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
15		difss 3254	. . . . . . . . 9 ⊢ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ 𝐴
16	14, 15	sstri 3157	. . . . . . . 8 ⊢ ∪ 𝐷 ⊆ 𝐴
17		sseq2 3172	. . . . . . . 8 ⊢ (dom 𝑓 = 𝐴 → (∪ 𝐷 ⊆ dom 𝑓 ↔ ∪ 𝐷 ⊆ 𝐴))
18	16, 17	mpbiri 167	. . . . . . 7 ⊢ (dom 𝑓 = 𝐴 → ∪ 𝐷 ⊆ dom 𝑓)
19		dfss 3136	. . . . . . 7 ⊢ (∪ 𝐷 ⊆ dom 𝑓 ↔ ∪ 𝐷 = (∪ 𝐷 ∩ dom 𝑓))
20	18, 19	sylib 121	. . . . . 6 ⊢ (dom 𝑓 = 𝐴 → ∪ 𝐷 = (∪ 𝐷 ∩ dom 𝑓))
21	20	uneq1d 3281	. . . . 5 ⊢ (dom 𝑓 = 𝐴 → (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)) = ((∪ 𝐷 ∩ dom 𝑓) ∪ (𝐴 ∖ ∪ 𝐷)))
22	12, 13	sbthlemi3 6940	. . . . . . . 8 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴) → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐴 ∖ ∪ 𝐷))
23		imassrn 4966	. . . . . . . 8 ⊢ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ ran 𝑔
24	22, 23	eqsstrrdi 3201	. . . . . . 7 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴) → (𝐴 ∖ ∪ 𝐷) ⊆ ran 𝑔)
25		dfss 3136	. . . . . . 7 ⊢ ((𝐴 ∖ ∪ 𝐷) ⊆ ran 𝑔 ↔ (𝐴 ∖ ∪ 𝐷) = ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔))
26	24, 25	sylib 121	. . . . . 6 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴) → (𝐴 ∖ ∪ 𝐷) = ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔))
27	26	uneq2d 3282	. . . . 5 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴) → ((∪ 𝐷 ∩ dom 𝑓) ∪ (𝐴 ∖ ∪ 𝐷)) = ((∪ 𝐷 ∩ dom 𝑓) ∪ ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔)))
28	21, 27	sylan9eq 2224	. . . 4 ⊢ ((dom 𝑓 = 𝐴 ∧ (EXMID ∧ ran 𝑔 ⊆ 𝐴)) → (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)) = ((∪ 𝐷 ∩ dom 𝑓) ∪ ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔)))
29	28	an12s 561	. . 3 ⊢ ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴)) → (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)) = ((∪ 𝐷 ∩ dom 𝑓) ∪ ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔)))
30	11, 29	eqtr4id 2223	. 2 ⊢ ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴)) → dom 𝐻 = (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)))
31		undifdcss 6904	. . . . 5 ⊢ (𝐴 = (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)) ↔ (∪ 𝐷 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ ∪ 𝐷))
32		exmidexmid 4183	. . . . . . 7 ⊢ (EXMID → DECID 𝑦 ∈ ∪ 𝐷)
33	32	ralrimivw 2545	. . . . . 6 ⊢ (EXMID → ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ ∪ 𝐷)
34	33	biantrud 302	. . . . 5 ⊢ (EXMID → (∪ 𝐷 ⊆ 𝐴 ↔ (∪ 𝐷 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ ∪ 𝐷)))
35	31, 34	bitr4id 198	. . . 4 ⊢ (EXMID → (𝐴 = (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)) ↔ ∪ 𝐷 ⊆ 𝐴))
36	16, 35	mpbiri 167	. . 3 ⊢ (EXMID → 𝐴 = (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)))
37	36	adantr 274	. 2 ⊢ ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴)) → 𝐴 = (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)))
38	30, 37	eqtr4d 2207	1 ⊢ ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴)) → dom 𝐻 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 103  DECID wdc 830   = wceq 1349   ∈ wcel 2142  {cab 2157  ∀wral 2449  Vcvv 2731   ∖ cdif 3119   ∪ cun 3120   ∩ cin 3121   ⊆ wss 3122  ∪ cuni 3797  EXMIDwem 4181  ^◡ccnv 4611  dom cdm 4612  ran crn 4613   ↾ cres 4614   “ cima 4615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-14 2145  ax-ext 2153  ax-sep 4108  ax-nul 4116  ax-pow 4161  ax-pr 4195

This theorem depends on definitions:  df-bi 116  df-stab 827  df-dc 831  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ral 2454  df-rex 2455  df-rab 2458  df-v 2733  df-dif 3124  df-un 3126  df-in 3128  df-ss 3135  df-nul 3416  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-br 3991  df-opab 4052  df-exmid 4182  df-xp 4618  df-cnv 4620  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625

This theorem is referenced by:  sbthlemi9  6946