Theorem sbthlem9 9023

Description: Lemma for sbth 9025. (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–1-1→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 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 9021	. . . . . . 7 ⊢ ((Fun 𝑓 ∧ Fun ◡𝑔) → Fun 𝐻)
5	1, 2, 3	sbthlem5 9019	. . . . . . . 8 ⊢ ((dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴) → dom 𝐻 = 𝐴)
6	5	adantrl 716	. . . . . . 7 ⊢ ((dom 𝑓 = 𝐴 ∧ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴)) → dom 𝐻 = 𝐴)
7	4, 6	anim12i 613	. . . . . 6 ⊢ (((Fun 𝑓 ∧ Fun ◡𝑔) ∧ (dom 𝑓 = 𝐴 ∧ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴))) → (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
8	7	an42s 661	. . . . 5 ⊢ (((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
9	8	adantlr 715	. . . 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 715	. . 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 9022	. . . 4 ⊢ ((Fun ◡𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → Fun ◡𝐻)
12	11	adantll 714	. . 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 484	. . . . . . 7 ⊢ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) → dom 𝑔 = 𝐵)
14	13	anim1i 615	. . . . . 6 ⊢ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) → (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴))
15		df-rn 5635	. . . . . . 7 ⊢ ran 𝐻 = dom ◡𝐻
16	1, 2, 3	sbthlem6 9020	. . . . . . 7 ⊢ ((ran 𝑓 ⊆ 𝐵 ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ran 𝐻 = 𝐵)
17	15, 16	eqtr3id 2785	. . . . . 6 ⊢ ((ran 𝑓 ⊆ 𝐵 ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → dom ◡𝐻 = 𝐵)
18	14, 17	sylanr1 682	. . . . 5 ⊢ ((ran 𝑓 ⊆ 𝐵 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → dom ◡𝐻 = 𝐵)
19	18	adantll 714	. . . 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 715	. . 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 515	. 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 6497	. . . 4 ⊢ (𝑓:𝐴–1-1→𝐵 ↔ (𝑓:𝐴⟶𝐵 ∧ Fun ◡𝑓))
23		df-f 6496	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓 ⊆ 𝐵))
24		df-fn 6495	. . . . . . 7 ⊢ (𝑓 Fn 𝐴 ↔ (Fun 𝑓 ∧ dom 𝑓 = 𝐴))
25	24	anbi1i 624	. . . . . 6 ⊢ ((𝑓 Fn 𝐴 ∧ ran 𝑓 ⊆ 𝐵) ↔ ((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓 ⊆ 𝐵))
26	23, 25	bitri 275	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 ↔ ((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓 ⊆ 𝐵))
27	26	anbi1i 624	. . . 4 ⊢ ((𝑓:𝐴⟶𝐵 ∧ Fun ◡𝑓) ↔ (((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓 ⊆ 𝐵) ∧ Fun ◡𝑓))
28	22, 27	bitri 275	. . 3 ⊢ (𝑓:𝐴–1-1→𝐵 ↔ (((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓 ⊆ 𝐵) ∧ Fun ◡𝑓))
29		df-f1 6497	. . . 4 ⊢ (𝑔:𝐵–1-1→𝐴 ↔ (𝑔:𝐵⟶𝐴 ∧ Fun ◡𝑔))
30		df-f 6496	. . . . . 6 ⊢ (𝑔:𝐵⟶𝐴 ↔ (𝑔 Fn 𝐵 ∧ ran 𝑔 ⊆ 𝐴))
31		df-fn 6495	. . . . . . 7 ⊢ (𝑔 Fn 𝐵 ↔ (Fun 𝑔 ∧ dom 𝑔 = 𝐵))
32	31	anbi1i 624	. . . . . 6 ⊢ ((𝑔 Fn 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ↔ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴))
33	30, 32	bitri 275	. . . . 5 ⊢ (𝑔:𝐵⟶𝐴 ↔ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴))
34	33	anbi1i 624	. . . 4 ⊢ ((𝑔:𝐵⟶𝐴 ∧ Fun ◡𝑔) ↔ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔))
35	29, 34	bitri 275	. . 3 ⊢ (𝑔:𝐵–1-1→𝐴 ↔ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔))
36	28, 35	anbi12i 628	. 2 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) ↔ ((((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓 ⊆ 𝐵) ∧ Fun ◡𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)))
37		dff1o4 6782	. . 3 ⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ (𝐻 Fn 𝐴 ∧ ◡𝐻 Fn 𝐵))
38		df-fn 6495	. . . 4 ⊢ (𝐻 Fn 𝐴 ↔ (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
39		df-fn 6495	. . . 4 ⊢ (◡𝐻 Fn 𝐵 ↔ (Fun ◡𝐻 ∧ dom ◡𝐻 = 𝐵))
40	38, 39	anbi12i 628	. . 3 ⊢ ((𝐻 Fn 𝐴 ∧ ◡𝐻 Fn 𝐵) ↔ ((Fun 𝐻 ∧ dom 𝐻 = 𝐴) ∧ (Fun ◡𝐻 ∧ dom ◡𝐻 = 𝐵)))
41	37, 40	bitri 275	. 2 ⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ ((Fun 𝐻 ∧ dom 𝐻 = 𝐴) ∧ (Fun ◡𝐻 ∧ dom ◡𝐻 = 𝐵)))
42	21, 36, 41	3imtr4i 292	1 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐻:𝐴–1-1-onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2714  Vcvv 3440   ∖ cdif 3898   ∪ cun 3899   ⊆ wss 3901  ∪ cuni 4863  ^◡ccnv 5623  dom cdm 5624  ran crn 5625   ↾ cres 5626   “ cima 5627  Fun wfun 6486   Fn wfn 6487  ⟶wf 6488  –1-1→wf1 6489  –1-1-onto→wf1o 6491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499

This theorem is referenced by:  sbthlem10  9024  sbthfilem  9122