Theorem sbthlemi6 7269

Description: Lemma for isbth 7274. (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 531	. . 3 ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → EXMID)
2		simprll 543	. . 3 ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → dom 𝑔 = 𝐵)
3		simprlr 544	. . 3 ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ran 𝑔 ⊆ 𝐴)
4		simprr 537	. . 3 ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → Fun ◡𝑔)
5		rnun 5191	. . . . 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 5005	. . . . 5 ⊢ ran 𝐻 = ran ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
8		df-ima 4782	. . . . . 6 ⊢ (𝑓 “ ∪ 𝐷) = ran (𝑓 ↾ ∪ 𝐷)
9	8	uneq1i 3379	. . . . 5 ⊢ ((𝑓 “ ∪ 𝐷) ∪ ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = (ran (𝑓 ↾ ∪ 𝐷) ∪ ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
10	5, 7, 9	3eqtr4i 2269	. . . 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 7267	. . . . . 6 ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
14		df-ima 4782	. . . . . 6 ⊢ (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))
15	13, 14	eqtr3di 2286	. . . . 5 ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
16	15	uneq2d 3383	. . . 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 2290	. . 3 ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → ran 𝐻 = ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
18	1, 2, 3, 4, 17	syl121anc 1283	. 2 ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ran 𝐻 = ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
19		imassrn 5132	. . . . . . 7 ⊢ (𝑓 “ ∪ 𝐷) ⊆ ran 𝑓
20		sstr2 3255	. . . . . . 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 7220	. . . . . . 7 ⊢ (𝐵 = ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ↔ ((𝑓 “ ∪ 𝐷) ⊆ 𝐵 ∧ ∀𝑦 ∈ 𝐵 DECID 𝑦 ∈ (𝑓 “ ∪ 𝐷)))
24		exmidexmid 4328	. . . . . . . . 9 ⊢ (EXMID → DECID 𝑦 ∈ (𝑓 “ ∪ 𝐷))
25	24	ralrimivw 2624	. . . . . . . 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 2244	. . 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 2271	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 846   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  Vcvv 2821   ∖ cdif 3217   ∪ cun 3218   ⊆ wss 3220  ∪ cuni 3930  EXMIDwem 4326  ^◡ccnv 4768  dom cdm 4769  ran crn 4770   ↾ cres 4771   “ cima 4772  Fun wfun 5366

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 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 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-exmid 4327  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-fun 5374

This theorem is referenced by:  sbthlemi9  7272