Theorem sbthlemi6 7021

Description: Lemma for isbth 7026. (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))}
sbthlem.3	⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))

Assertion

Ref	Expression
sbthlemi6	⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ran 𝐻 = 𝐵)

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 sbthlemi6

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 527	. . 3 ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → EXMID)
2		simprll 537	. . 3 ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → dom 𝑔 = 𝐵)
3		simprlr 538	. . 3 ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ran 𝑔 ⊆ 𝐴)
4		simprr 531	. . 3 ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → Fun ◡𝑔)
5		rnun 5074	. . . . 5 ⊢ ran ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = (ran (𝑓 ↾ ∪ 𝐷) ∪ ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
6		sbthlem.3	. . . . . 6 ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
7	6	rneqi 4890	. . . . 5 ⊢ ran 𝐻 = ran ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
8		df-ima 4672	. . . . . 6 ⊢ (𝑓 “ ∪ 𝐷) = ran (𝑓 ↾ ∪ 𝐷)
9	8	uneq1i 3309	. . . . 5 ⊢ ((𝑓 “ ∪ 𝐷) ∪ ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = (ran (𝑓 ↾ ∪ 𝐷) ∪ ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
10	5, 7, 9	3eqtr4i 2224	. . . 4 ⊢ ran 𝐻 = ((𝑓 “ ∪ 𝐷) ∪ ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
11		sbthlem.1	. . . . . . 7 ⊢ 𝐴 ∈ V
12		sbthlem.2	. . . . . . 7 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}
13	11, 12	sbthlemi4 7019	. . . . . 6 ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
14		df-ima 4672	. . . . . 6 ⊢ (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))
15	13, 14	eqtr3di 2241	. . . . 5 ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
16	15	uneq2d 3313	. . . 4 ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = ((𝑓 “ ∪ 𝐷) ∪ ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))))
17	10, 16	eqtr4id 2245	. . 3 ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → ran 𝐻 = ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
18	1, 2, 3, 4, 17	syl121anc 1254	. 2 ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ran 𝐻 = ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
19		imassrn 5016	. . . . . . 7 ⊢ (𝑓 “ ∪ 𝐷) ⊆ ran 𝑓
20		sstr2 3186	. . . . . . 7 ⊢ ((𝑓 “ ∪ 𝐷) ⊆ ran 𝑓 → (ran 𝑓 ⊆ 𝐵 → (𝑓 “ ∪ 𝐷) ⊆ 𝐵))
21	19, 20	ax-mp 5	. . . . . 6 ⊢ (ran 𝑓 ⊆ 𝐵 → (𝑓 “ ∪ 𝐷) ⊆ 𝐵)
22	21	adantl 277	. . . . 5 ⊢ ((EXMID ∧ ran 𝑓 ⊆ 𝐵) → (𝑓 “ ∪ 𝐷) ⊆ 𝐵)
23		undifdcss 6979	. . . . . . 7 ⊢ (𝐵 = ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ↔ ((𝑓 “ ∪ 𝐷) ⊆ 𝐵 ∧ ∀𝑦 ∈ 𝐵 DECID 𝑦 ∈ (𝑓 “ ∪ 𝐷)))
24		exmidexmid 4225	. . . . . . . . 9 ⊢ (EXMID → DECID 𝑦 ∈ (𝑓 “ ∪ 𝐷))
25	24	ralrimivw 2568	. . . . . . . 8 ⊢ (EXMID → ∀𝑦 ∈ 𝐵 DECID 𝑦 ∈ (𝑓 “ ∪ 𝐷))
26	25	biantrud 304	. . . . . . 7 ⊢ (EXMID → ((𝑓 “ ∪ 𝐷) ⊆ 𝐵 ↔ ((𝑓 “ ∪ 𝐷) ⊆ 𝐵 ∧ ∀𝑦 ∈ 𝐵 DECID 𝑦 ∈ (𝑓 “ ∪ 𝐷))))
27	23, 26	bitr4id 199	. . . . . 6 ⊢ (EXMID → (𝐵 = ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ↔ (𝑓 “ ∪ 𝐷) ⊆ 𝐵))
28	27	adantr 276	. . . . 5 ⊢ ((EXMID ∧ ran 𝑓 ⊆ 𝐵) → (𝐵 = ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ↔ (𝑓 “ ∪ 𝐷) ⊆ 𝐵))
29	22, 28	mpbird 167	. . . 4 ⊢ ((EXMID ∧ ran 𝑓 ⊆ 𝐵) → 𝐵 = ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
30	29	eqcomd 2199	. . 3 ⊢ ((EXMID ∧ ran 𝑓 ⊆ 𝐵) → ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = 𝐵)
31	30	adantr 276	. 2 ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = 𝐵)
32	18, 31	eqtrd 2226	1 ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ran 𝐻 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 835   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  {cab 2179  ∀wral 2472  Vcvv 2760   ∖ cdif 3150   ∪ cun 3151   ⊆ wss 3153  ∪ cuni 3835  EXMIDwem 4223  ^◡ccnv 4658  dom cdm 4659  ran crn 4660   ↾ cres 4661   “ cima 4662  Fun wfun 5248

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-exmid 4224  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-fun 5256

This theorem is referenced by:  sbthlemi9  7024