Theorem sbthlemi9 7026

Description: Lemma for isbth 7028. (Contributed by NM, 28-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
sbthlemi9	⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐻:𝐴–1-1-onto→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 sbthlemi9

Step	Hyp	Ref	Expression
1		simp2 1000	. . . . . . . . . 10 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝑓:𝐴–1-1→𝐵)
2		df-f1 5260	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1→𝐵 ↔ (𝑓:𝐴⟶𝐵 ∧ Fun ◡𝑓))
3	1, 2	sylib 122	. . . . . . . . 9 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (𝑓:𝐴⟶𝐵 ∧ Fun ◡𝑓))
4	3	simpld 112	. . . . . . . 8 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝑓:𝐴⟶𝐵)
5		df-f 5259	. . . . . . . 8 ⊢ (𝑓:𝐴⟶𝐵 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓 ⊆ 𝐵))
6	4, 5	sylib 122	. . . . . . 7 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (𝑓 Fn 𝐴 ∧ ran 𝑓 ⊆ 𝐵))
7	6	simpld 112	. . . . . 6 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝑓 Fn 𝐴)
8		df-fn 5258	. . . . . 6 ⊢ (𝑓 Fn 𝐴 ↔ (Fun 𝑓 ∧ dom 𝑓 = 𝐴))
9	7, 8	sylib 122	. . . . 5 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (Fun 𝑓 ∧ dom 𝑓 = 𝐴))
10	9	simpld 112	. . . 4 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → Fun 𝑓)
11		simp3 1001	. . . . . 6 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝑔:𝐵–1-1→𝐴)
12		df-f1 5260	. . . . . 6 ⊢ (𝑔:𝐵–1-1→𝐴 ↔ (𝑔:𝐵⟶𝐴 ∧ Fun ◡𝑔))
13	11, 12	sylib 122	. . . . 5 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (𝑔:𝐵⟶𝐴 ∧ Fun ◡𝑔))
14	13	simprd 114	. . . 4 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → Fun ◡𝑔)
15		sbthlem.1	. . . . 5 ⊢ 𝐴 ∈ V
16		sbthlem.2	. . . . 5 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}
17		sbthlem.3	. . . . 5 ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
18	15, 16, 17	sbthlem7 7024	. . . 4 ⊢ ((Fun 𝑓 ∧ Fun ◡𝑔) → Fun 𝐻)
19	10, 14, 18	syl2anc 411	. . 3 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → Fun 𝐻)
20		simp1 999	. . . 4 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → EXMID)
21	9	simprd 114	. . . 4 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → dom 𝑓 = 𝐴)
22	13	simpld 112	. . . . . 6 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝑔:𝐵⟶𝐴)
23		df-f 5259	. . . . . 6 ⊢ (𝑔:𝐵⟶𝐴 ↔ (𝑔 Fn 𝐵 ∧ ran 𝑔 ⊆ 𝐴))
24	22, 23	sylib 122	. . . . 5 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (𝑔 Fn 𝐵 ∧ ran 𝑔 ⊆ 𝐴))
25	24	simprd 114	. . . 4 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → ran 𝑔 ⊆ 𝐴)
26	15, 16, 17	sbthlemi5 7022	. . . 4 ⊢ ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴)) → dom 𝐻 = 𝐴)
27	20, 21, 25, 26	syl12anc 1247	. . 3 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → dom 𝐻 = 𝐴)
28		df-fn 5258	. . 3 ⊢ (𝐻 Fn 𝐴 ↔ (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
29	19, 27, 28	sylanbrc 417	. 2 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐻 Fn 𝐴)
30	3	simprd 114	. . . 4 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → Fun ◡𝑓)
31	24	simpld 112	. . . . . 6 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝑔 Fn 𝐵)
32		df-fn 5258	. . . . . 6 ⊢ (𝑔 Fn 𝐵 ↔ (Fun 𝑔 ∧ dom 𝑔 = 𝐵))
33	31, 32	sylib 122	. . . . 5 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (Fun 𝑔 ∧ dom 𝑔 = 𝐵))
34	33, 25	jca 306	. . . 4 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴))
35	15, 16, 17	sbthlemi8 7025	. . . 4 ⊢ (((EXMID ∧ Fun ◡𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → Fun ◡𝐻)
36	20, 30, 34, 14, 35	syl22anc 1250	. . 3 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → Fun ◡𝐻)
37	6	simprd 114	. . . 4 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → ran 𝑓 ⊆ 𝐵)
38	33	simprd 114	. . . . 5 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → dom 𝑔 = 𝐵)
39	38, 25	jca 306	. . . 4 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴))
40		df-rn 4671	. . . . 5 ⊢ ran 𝐻 = dom ◡𝐻
41	15, 16, 17	sbthlemi6 7023	. . . . 5 ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ran 𝐻 = 𝐵)
42	40, 41	eqtr3id 2240	. . . 4 ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → dom ◡𝐻 = 𝐵)
43	20, 37, 39, 14, 42	syl22anc 1250	. . 3 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → dom ◡𝐻 = 𝐵)
44		df-fn 5258	. . 3 ⊢ (◡𝐻 Fn 𝐵 ↔ (Fun ◡𝐻 ∧ dom ◡𝐻 = 𝐵))
45	36, 43, 44	sylanbrc 417	. 2 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → ◡𝐻 Fn 𝐵)
46		dff1o4 5509	. 2 ⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ (𝐻 Fn 𝐴 ∧ ◡𝐻 Fn 𝐵))
47	29, 45, 46	sylanbrc 417	1 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐻:𝐴–1-1-onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  {cab 2179  Vcvv 2760   ∖ cdif 3151   ∪ cun 3152   ⊆ wss 3154  ∪ cuni 3836  EXMIDwem 4224  ^◡ccnv 4659  dom cdm 4660  ran crn 4661   ↾ cres 4662   “ cima 4663  Fun wfun 5249   Fn wfn 5250  ⟶wf 5251  –1-1→wf1 5252  –1-1-onto→wf1o 5254

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 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239

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 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-exmid 4225  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262

This theorem is referenced by:  sbthlemi10  7027