Theorem sbthlem9 9152

Description: Lemma for sbth 9154. (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
sbthlem9	⊢ ((𝑓:𝐴⟶1-1𝐵 ∧ 𝑔:𝐵⟶1-1𝐴) → 𝐻:𝐴⟶1-1onto𝐵)

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 sbthlem9

Step	Hyp	Ref	Expression
1		sbthlem.1	. . . . . . . 8 ⊢ 𝐴 ∈ V
2		sbthlem.2	. . . . . . . 8 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}
3		sbthlem.3	. . . . . . . 8 ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
4	1, 2, 3	sbthlem7 9150	. . . . . . 7 ⊢ ((Fun 𝑓 ∧ Fun ◡𝑔) → Fun 𝐻)
5	1, 2, 3	sbthlem5 9148	. . . . . . . 8 ⊢ ((dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴) → dom 𝐻 = 𝐴)
6	5	adantrl 729	. . . . . . 7 ⊢ ((dom 𝑓 = 𝐴 ∧ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴)) → dom 𝐻 = 𝐴)
7	4, 6	anim12i 625	. . . . . 6 ⊢ (((Fun 𝑓 ∧ Fun ◡𝑔) ∧ (dom 𝑓 = 𝐴 ∧ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴))) → (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
8	7	an42s 674	. . . . 5 ⊢ (((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
9	8	adantlr 728	. . . 4 ⊢ ((((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓 ⊆ 𝐵) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
10	9	adantlr 728	. . 3 ⊢ (((((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓 ⊆ 𝐵) ∧ Fun ◡𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
11	1, 2, 3	sbthlem8 9151	. . . 4 ⊢ ((Fun ◡𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → Fun ◡𝐻)
12	11	adantll 727	. . 3 ⊢ (((((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓 ⊆ 𝐵) ∧ Fun ◡𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → Fun ◡𝐻)
13		simpr 490	. . . . . . 7 ⊢ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) → dom 𝑔 = 𝐵)
14	13	anim1i 627	. . . . . 6 ⊢ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) → (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴))
15		df-rn 5669	. . . . . . 7 ⊢ ran 𝐻 = dom ◡𝐻
16	1, 2, 3	sbthlem6 9149	. . . . . . 7 ⊢ ((ran 𝑓 ⊆ 𝐵 ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ran 𝐻 = 𝐵)
17	15, 16	eqtr3id 2812	. . . . . 6 ⊢ ((ran 𝑓 ⊆ 𝐵 ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → dom ◡𝐻 = 𝐵)
18	14, 17	sylanr1 695	. . . . 5 ⊢ ((ran 𝑓 ⊆ 𝐵 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → dom ◡𝐻 = 𝐵)
19	18	adantll 727	. . . 4 ⊢ ((((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓 ⊆ 𝐵) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → dom ◡𝐻 = 𝐵)
20	19	adantlr 728	. . 3 ⊢ (((((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓 ⊆ 𝐵) ∧ Fun ◡𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → dom ◡𝐻 = 𝐵)
21	10, 12, 20	jca32 525	. 2 ⊢ (((((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓 ⊆ 𝐵) ∧ Fun ◡𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ((Fun 𝐻 ∧ dom 𝐻 = 𝐴) ∧ (Fun ◡𝐻 ∧ dom ◡𝐻 = 𝐵)))
22		df-f1 6571	. . . 4 ⊢ (𝑓:𝐴⟶1-1𝐵 ↔ (𝑓:𝐴⟶𝐵 ∧ Fun ◡𝑓))
23		df-f 6570	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓 ⊆ 𝐵))
24		df-fn 6569	. . . . . . 7 ⊢ (𝑓 Fn 𝐴 ↔ (Fun 𝑓 ∧ dom 𝑓 = 𝐴))
25	24	anbi1i 636	. . . . . 6 ⊢ ((𝑓 Fn 𝐴 ∧ ran 𝑓 ⊆ 𝐵) ↔ ((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓 ⊆ 𝐵))
26	23, 25	bitri 278	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 ↔ ((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓 ⊆ 𝐵))
27	26	anbi1i 636	. . . 4 ⊢ ((𝑓:𝐴⟶𝐵 ∧ Fun ◡𝑓) ↔ (((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓 ⊆ 𝐵) ∧ Fun ◡𝑓))
28	22, 27	bitri 278	. . 3 ⊢ (𝑓:𝐴⟶1-1𝐵 ↔ (((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓 ⊆ 𝐵) ∧ Fun ◡𝑓))
29		df-f1 6571	. . . 4 ⊢ (𝑔:𝐵⟶1-1𝐴 ↔ (𝑔:𝐵⟶𝐴 ∧ Fun ◡𝑔))
30		df-f 6570	. . . . . 6 ⊢ (𝑔:𝐵⟶𝐴 ↔ (𝑔 Fn 𝐵 ∧ ran 𝑔 ⊆ 𝐴))
31		df-fn 6569	. . . . . . 7 ⊢ (𝑔 Fn 𝐵 ↔ (Fun 𝑔 ∧ dom 𝑔 = 𝐵))
32	31	anbi1i 636	. . . . . 6 ⊢ ((𝑔 Fn 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ↔ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴))
33	30, 32	bitri 278	. . . . 5 ⊢ (𝑔:𝐵⟶𝐴 ↔ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴))
34	33	anbi1i 636	. . . 4 ⊢ ((𝑔:𝐵⟶𝐴 ∧ Fun ◡𝑔) ↔ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔))
35	29, 34	bitri 278	. . 3 ⊢ (𝑔:𝐵⟶1-1𝐴 ↔ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔))
36	28, 35	anbi12i 640	. 2 ⊢ ((𝑓:𝐴⟶1-1𝐵 ∧ 𝑔:𝐵⟶1-1𝐴) ↔ ((((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓 ⊆ 𝐵) ∧ Fun ◡𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)))
37		dff1o4 6863	. . 3 ⊢ (𝐻:𝐴⟶1-1onto𝐵 ↔ (𝐻 Fn 𝐴 ∧ ◡𝐻 Fn 𝐵))
38		df-fn 6569	. . . 4 ⊢ (𝐻 Fn 𝐴 ↔ (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
39		df-fn 6569	. . . 4 ⊢ (◡𝐻 Fn 𝐵 ↔ (Fun ◡𝐻 ∧ dom ◡𝐻 = 𝐵))
40	38, 39	anbi12i 640	. . 3 ⊢ ((𝐻 Fn 𝐴 ∧ ◡𝐻 Fn 𝐵) ↔ ((Fun 𝐻 ∧ dom 𝐻 = 𝐴) ∧ (Fun ◡𝐻 ∧ dom ◡𝐻 = 𝐵)))
41	37, 40	bitri 278	. 2 ⊢ (𝐻:𝐴⟶1-1onto𝐵 ↔ ((Fun 𝐻 ∧ dom 𝐻 = 𝐴) ∧ (Fun ◡𝐻 ∧ dom ◡𝐻 = 𝐵)))
42	21, 36, 41	3imtr4i 295	1 ⊢ ((𝑓:𝐴⟶1-1𝐵 ∧ 𝑔:𝐵⟶1-1𝐴) → 𝐻:𝐴⟶1-1onto𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   ∪ cun 4152  ∪ cuni 4872  ^◡ccnv 5657  dom cdm 5658  ran crn 5659   ↾ cres 5660   “ cima 5661  Fun wfun 6560   Fn wfn 6561  ⟶wf 6562  ⟶1-1wf1 6563  ⟶1-1ontowf1o 6565

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573

This theorem is used by:  sbthlem10  9153  sbthfilem  9251