Theorem sbthlemi6 6939

Description: Lemma for isbth 6944. (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 524	. . 3 ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → EXMID)
2		simprll 532	. . 3 ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → dom 𝑔 = 𝐵)
3		simprlr 533	. . 3 ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ran 𝑔 ⊆ 𝐴)
4		simprr 527	. . 3 ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → Fun ◡𝑔)
5		rnun 5019	. . . . 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 4839	. . . . 5 ⊢ ran 𝐻 = ran ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
8		df-ima 4624	. . . . . 6 ⊢ (𝑓 “ ∪ 𝐷) = ran (𝑓 ↾ ∪ 𝐷)
9	8	uneq1i 3277	. . . . 5 ⊢ ((𝑓 “ ∪ 𝐷) ∪ ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = (ran (𝑓 ↾ ∪ 𝐷) ∪ ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
10	5, 7, 9	3eqtr4i 2201	. . . 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 6937	. . . . . 6 ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
14		df-ima 4624	. . . . . 6 ⊢ (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))
15	13, 14	eqtr3di 2218	. . . . 5 ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
16	15	uneq2d 3281	. . . 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 2222	. . 3 ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → ran 𝐻 = ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
18	1, 2, 3, 4, 17	syl121anc 1238	. 2 ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ran 𝐻 = ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
19		imassrn 4964	. . . . . . 7 ⊢ (𝑓 “ ∪ 𝐷) ⊆ ran 𝑓
20		sstr2 3154	. . . . . . 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 275	. . . . 5 ⊢ ((EXMID ∧ ran 𝑓 ⊆ 𝐵) → (𝑓 “ ∪ 𝐷) ⊆ 𝐵)
23		undifdcss 6900	. . . . . . 7 ⊢ (𝐵 = ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ↔ ((𝑓 “ ∪ 𝐷) ⊆ 𝐵 ∧ ∀𝑦 ∈ 𝐵 DECID 𝑦 ∈ (𝑓 “ ∪ 𝐷)))
24		exmidexmid 4182	. . . . . . . . 9 ⊢ (EXMID → DECID 𝑦 ∈ (𝑓 “ ∪ 𝐷))
25	24	ralrimivw 2544	. . . . . . . 8 ⊢ (EXMID → ∀𝑦 ∈ 𝐵 DECID 𝑦 ∈ (𝑓 “ ∪ 𝐷))
26	25	biantrud 302	. . . . . . 7 ⊢ (EXMID → ((𝑓 “ ∪ 𝐷) ⊆ 𝐵 ↔ ((𝑓 “ ∪ 𝐷) ⊆ 𝐵 ∧ ∀𝑦 ∈ 𝐵 DECID 𝑦 ∈ (𝑓 “ ∪ 𝐷))))
27	23, 26	bitr4id 198	. . . . . 6 ⊢ (EXMID → (𝐵 = ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ↔ (𝑓 “ ∪ 𝐷) ⊆ 𝐵))
28	27	adantr 274	. . . . 5 ⊢ ((EXMID ∧ ran 𝑓 ⊆ 𝐵) → (𝐵 = ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) ↔ (𝑓 “ ∪ 𝐷) ⊆ 𝐵))
29	22, 28	mpbird 166	. . . 4 ⊢ ((EXMID ∧ ran 𝑓 ⊆ 𝐵) → 𝐵 = ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
30	29	eqcomd 2176	. . 3 ⊢ ((EXMID ∧ ran 𝑓 ⊆ 𝐵) → ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = 𝐵)
31	30	adantr 274	. 2 ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = 𝐵)
32	18, 31	eqtrd 2203	1 ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ran 𝐻 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 829   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  {cab 2156  ∀wral 2448  Vcvv 2730   ∖ cdif 3118   ∪ cun 3119   ⊆ wss 3121  ∪ cuni 3796  EXMIDwem 4180  ^◡ccnv 4610  dom cdm 4611  ran crn 4612   ↾ cres 4613   “ cima 4614  Fun wfun 5192

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-exmid 4181  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-fun 5200

This theorem is referenced by:  sbthlemi9  6942