Theorem sbthlem5 9019

Description: Lemma for sbth 9025. (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
sbthlem5	⊢ ((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 sbthlem5

Step	Hyp	Ref	Expression
1		sbthlem.3	. . . . 5 ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
2	1	dmeqi 5846	. . . 4 ⊢ dom 𝐻 = dom ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
3		dmun 5852	. . . . 5 ⊢ dom ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = (dom (𝑓 ↾ ∪ 𝐷) ∪ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
4		dmres 5964	. . . . . 6 ⊢ dom (𝑓 ↾ ∪ 𝐷) = (∪ 𝐷 ∩ dom 𝑓)
5		dmres 5964	. . . . . . 7 ⊢ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) = ((𝐴 ∖ ∪ 𝐷) ∩ dom ◡𝑔)
6		df-rn 5629	. . . . . . . . 9 ⊢ ran 𝑔 = dom ◡𝑔
7	6	eqcomi 2748	. . . . . . . 8 ⊢ dom ◡𝑔 = ran 𝑔
8	7	ineq2i 4146	. . . . . . 7 ⊢ ((𝐴 ∖ ∪ 𝐷) ∩ dom ◡𝑔) = ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔)
9	5, 8	eqtri 2762	. . . . . 6 ⊢ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) = ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔)
10	4, 9	uneq12i 4096	. . . . 5 ⊢ (dom (𝑓 ↾ ∪ 𝐷) ∪ dom (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ((∪ 𝐷 ∩ dom 𝑓) ∪ ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔))
11	3, 10	eqtri 2762	. . . 4 ⊢ dom ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ((∪ 𝐷 ∩ dom 𝑓) ∪ ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔))
12	2, 11	eqtri 2762	. . 3 ⊢ dom 𝐻 = ((∪ 𝐷 ∩ dom 𝑓) ∪ ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔))
13		sbthlem.1	. . . . . . . . 9 ⊢ 𝐴 ∈ V
14		sbthlem.2	. . . . . . . . 9 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}
15	13, 14	sbthlem1 9015	. . . . . . . 8 ⊢ ∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
16		difss 4066	. . . . . . . 8 ⊢ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ 𝐴
17	15, 16	sstri 3924	. . . . . . 7 ⊢ ∪ 𝐷 ⊆ 𝐴
18		sseq2 3941	. . . . . . 7 ⊢ (dom 𝑓 = 𝐴 → (∪ 𝐷 ⊆ dom 𝑓 ↔ ∪ 𝐷 ⊆ 𝐴))
19	17, 18	mpbiri 259	. . . . . 6 ⊢ (dom 𝑓 = 𝐴 → ∪ 𝐷 ⊆ dom 𝑓)
20		dfss 3902	. . . . . 6 ⊢ (∪ 𝐷 ⊆ dom 𝑓 ↔ ∪ 𝐷 = (∪ 𝐷 ∩ dom 𝑓))
21	19, 20	sylib 219	. . . . 5 ⊢ (dom 𝑓 = 𝐴 → ∪ 𝐷 = (∪ 𝐷 ∩ dom 𝑓))
22	21	uneq1d 4097	. . . 4 ⊢ (dom 𝑓 = 𝐴 → (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)) = ((∪ 𝐷 ∩ dom 𝑓) ∪ (𝐴 ∖ ∪ 𝐷)))
23	13, 14	sbthlem3 9017	. . . . . . 7 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐴 ∖ ∪ 𝐷))
24		imassrn 6023	. . . . . . 7 ⊢ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ⊆ ran 𝑔
25	23, 24	eqsstrrdi 3960	. . . . . 6 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝐴 ∖ ∪ 𝐷) ⊆ ran 𝑔)
26		dfss 3902	. . . . . 6 ⊢ ((𝐴 ∖ ∪ 𝐷) ⊆ ran 𝑔 ↔ (𝐴 ∖ ∪ 𝐷) = ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔))
27	25, 26	sylib 219	. . . . 5 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝐴 ∖ ∪ 𝐷) = ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔))
28	27	uneq2d 4098	. . . 4 ⊢ (ran 𝑔 ⊆ 𝐴 → ((∪ 𝐷 ∩ dom 𝑓) ∪ (𝐴 ∖ ∪ 𝐷)) = ((∪ 𝐷 ∩ dom 𝑓) ∪ ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔)))
29	22, 28	sylan9eq 2794	. . 3 ⊢ ((dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴) → (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)) = ((∪ 𝐷 ∩ dom 𝑓) ∪ ((𝐴 ∖ ∪ 𝐷) ∩ ran 𝑔)))
30	12, 29	eqtr4id 2793	. 2 ⊢ ((dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴) → dom 𝐻 = (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)))
31		undif 4410	. . 3 ⊢ (∪ 𝐷 ⊆ 𝐴 ↔ (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)) = 𝐴)
32	17, 31	mpbi 231	. 2 ⊢ (∪ 𝐷 ∪ (𝐴 ∖ ∪ 𝐷)) = 𝐴
33	30, 32	eqtrdi 2790	1 ⊢ ((dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴) → dom 𝐻 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2717  Vcvv 3431   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∪ cuni 4838  ^◡ccnv 5617  dom cdm 5618  ran crn 5619   ↾ cres 5620   “ cima 5621

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-xp 5624  df-cnv 5626  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631

This theorem is referenced by:  sbthlem9  9023