Theorem sbthlemi4 7150

Description: Lemma for isbth 7157. (Contributed by NM, 27-Mar-1998.)

Hypotheses

Ref	Expression
sbthlem.1	⊢ 𝐴 ∈ V
sbthlem.2	⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}

Assertion

Ref	Expression
sbthlemi4	⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 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 sbthlemi4

Step	Hyp	Ref	Expression
1		df-ima 4736	. 2 ⊢ (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))
2		difss 3331	. . . . . . . 8 ⊢ (𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ 𝐵
3		sseq2 3249	. . . . . . . 8 ⊢ (dom 𝑔 = 𝐵 → ((𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ dom 𝑔 ↔ (𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ 𝐵))
4	2, 3	mpbiri 168	. . . . . . 7 ⊢ (dom 𝑔 = 𝐵 → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ dom 𝑔)
5		ssdmres 5033	. . . . . . 7 ⊢ ((𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ dom 𝑔 ↔ dom (𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
6	4, 5	sylib 122	. . . . . 6 ⊢ (dom 𝑔 = 𝐵 → dom (𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
7		dfdm4 4921	. . . . . 6 ⊢ dom (𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
8	6, 7	eqtr3di 2277	. . . . 5 ⊢ (dom 𝑔 = 𝐵 → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) = ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
9	8	adantr 276	. . . 4 ⊢ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) = ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
10	9	3ad2ant2 1043	. . 3 ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) = ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
11		funcnvres 5400	. . . . . . 7 ⊢ (Fun ◡𝑔 → ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (◡𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
12	11	3ad2ant3 1044	. . . . . 6 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴 ∧ Fun ◡𝑔) → ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (◡𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
13		sbthlem.1	. . . . . . . . 9 ⊢ 𝐴 ∈ V
14		sbthlem.2	. . . . . . . . 9 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}
15	13, 14	sbthlemi3 7149	. . . . . . . 8 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴) → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐴 ∖ ∪ 𝐷))
16	15	reseq2d 5011	. . . . . . 7 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴) → (◡𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) = (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
17	16	3adant3 1041	. . . . . 6 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴 ∧ Fun ◡𝑔) → (◡𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) = (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
18	12, 17	eqtrd 2262	. . . . 5 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴 ∧ Fun ◡𝑔) → ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
19	18	rneqd 4959	. . . 4 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴 ∧ Fun ◡𝑔) → ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
20	19	3adant2l 1256	. . 3 ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
21	10, 20	eqtrd 2262	. 2 ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
22	1, 21	eqtr4id 2281	1 ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  {cab 2215  Vcvv 2800   ∖ cdif 3195   ⊆ wss 3198  ∪ cuni 3891  EXMIDwem 4282  ^◡ccnv 4722  dom cdm 4723  ran crn 4724   ↾ cres 4725   “ cima 4726  Fun wfun 5318

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-exmid 4283  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-fun 5326

This theorem is referenced by:  sbthlemi6  7152  sbthlemi8  7154