Theorem sbthlem9 9087

Description: Lemma for sbth 9089. (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 9085	. . . . . . 7 ⊢ ((Fun 𝑓 ∧ Fun ◡𝑔) → Fun 𝐻)
5	1, 2, 3	sbthlem5 9083	. . . . . . . 8 ⊢ ((dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴) → dom 𝐻 = 𝐴)
6	5	adantrl 714	. . . . . . 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 659	. . . . 5 ⊢ (((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
9	8	adantlr 713	. . . 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 713	. . 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 9086	. . . 4 ⊢ ((Fun ◡𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → Fun ◡𝐻)
12	11	adantll 712	. . 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 485	. . . . . . 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 5686	. . . . . . 7 ⊢ ran 𝐻 = dom ◡𝐻
16	1, 2, 3	sbthlem6 9084	. . . . . . 7 ⊢ ((ran 𝑓 ⊆ 𝐵 ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ran 𝐻 = 𝐵)
17	15, 16	eqtr3id 2786	. . . . . 6 ⊢ ((ran 𝑓 ⊆ 𝐵 ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → dom ◡𝐻 = 𝐵)
18	14, 17	sylanr1 680	. . . . 5 ⊢ ((ran 𝑓 ⊆ 𝐵 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → dom ◡𝐻 = 𝐵)
19	18	adantll 712	. . . 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 713	. . 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 516	. 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 6545	. . . 4 ⊢ (𝑓:𝐴–1-1→𝐵 ↔ (𝑓:𝐴⟶𝐵 ∧ Fun ◡𝑓))
23		df-f 6544	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓 ⊆ 𝐵))
24		df-fn 6543	. . . . . . 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 274	. . . . 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 274	. . 3 ⊢ (𝑓:𝐴–1-1→𝐵 ↔ (((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓 ⊆ 𝐵) ∧ Fun ◡𝑓))
29		df-f1 6545	. . . 4 ⊢ (𝑔:𝐵–1-1→𝐴 ↔ (𝑔:𝐵⟶𝐴 ∧ Fun ◡𝑔))
30		df-f 6544	. . . . . 6 ⊢ (𝑔:𝐵⟶𝐴 ↔ (𝑔 Fn 𝐵 ∧ ran 𝑔 ⊆ 𝐴))
31		df-fn 6543	. . . . . . 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 274	. . . . 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 274	. . 3 ⊢ (𝑔:𝐵–1-1→𝐴 ↔ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔))
36	28, 35	anbi12i 627	. 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 6838	. . 3 ⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ (𝐻 Fn 𝐴 ∧ ◡𝐻 Fn 𝐵))
38		df-fn 6543	. . . 4 ⊢ (𝐻 Fn 𝐴 ↔ (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
39		df-fn 6543	. . . 4 ⊢ (◡𝐻 Fn 𝐵 ↔ (Fun ◡𝐻 ∧ dom ◡𝐻 = 𝐵))
40	38, 39	anbi12i 627	. . 3 ⊢ ((𝐻 Fn 𝐴 ∧ ◡𝐻 Fn 𝐵) ↔ ((Fun 𝐻 ∧ dom 𝐻 = 𝐴) ∧ (Fun ◡𝐻 ∧ dom ◡𝐻 = 𝐵)))
41	37, 40	bitri 274	. 2 ⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ ((Fun 𝐻 ∧ dom 𝐻 = 𝐴) ∧ (Fun ◡𝐻 ∧ dom ◡𝐻 = 𝐵)))
42	21, 36, 41	3imtr4i 291	1 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐻:𝐴–1-1-onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2709  Vcvv 3474   ∖ cdif 3944   ∪ cun 3945   ⊆ wss 3947  ∪ cuni 4907  ^◡ccnv 5674  dom cdm 5675  ran crn 5676   ↾ cres 5677   “ cima 5678  Fun wfun 6534   Fn wfn 6535  ⟶wf 6536  –1-1→wf1 6537  –1-1-onto→wf1o 6539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547

This theorem is referenced by:  sbthlem10  9088  sbthfilem  9197