Theorem sbthlem10 9088

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)))
sbthlem.4	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
sbthlem10	⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝑓,𝑔   𝑥,𝐻   𝑓,𝑔,𝐴   𝐵,𝑓,𝑔

Allowed substitution hints:   𝐷(𝑓,𝑔)   𝐻(𝑓,𝑔)

Proof of Theorem sbthlem10

Step	Hyp	Ref	Expression
1		sbthlem.4	. . . . 5 ⊢ 𝐵 ∈ V
2	1	brdom 8952	. . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)
3		sbthlem.1	. . . . 5 ⊢ 𝐴 ∈ V
4	3	brdom 8952	. . . 4 ⊢ (𝐵 ≼ 𝐴 ↔ ∃𝑔 𝑔:𝐵–1-1→𝐴)
5	2, 4	anbi12i 627	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) ↔ (∃𝑓 𝑓:𝐴–1-1→𝐵 ∧ ∃𝑔 𝑔:𝐵–1-1→𝐴))
6		exdistrv 1959	. . 3 ⊢ (∃𝑓∃𝑔(𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) ↔ (∃𝑓 𝑓:𝐴–1-1→𝐵 ∧ ∃𝑔 𝑔:𝐵–1-1→𝐴))
7	5, 6	bitr4i 277	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) ↔ ∃𝑓∃𝑔(𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴))
8		sbthlem.3	. . . . 5 ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
9		vex 3478	. . . . . . 7 ⊢ 𝑓 ∈ V
10	9	resex 6027	. . . . . 6 ⊢ (𝑓 ↾ ∪ 𝐷) ∈ V
11		vex 3478	. . . . . . . 8 ⊢ 𝑔 ∈ V
12	11	cnvex 7912	. . . . . . 7 ⊢ ◡𝑔 ∈ V
13	12	resex 6027	. . . . . 6 ⊢ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) ∈ V
14	10, 13	unex 7729	. . . . 5 ⊢ ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ∈ V
15	8, 14	eqeltri 2829	. . . 4 ⊢ 𝐻 ∈ V
16		sbthlem.2	. . . . 5 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}
17	3, 16, 8	sbthlem9 9087	. . . 4 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐻:𝐴–1-1-onto→𝐵)
18		f1oen3g 8958	. . . 4 ⊢ ((𝐻 ∈ V ∧ 𝐻:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
19	15, 17, 18	sylancr 587	. . 3 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐴 ≈ 𝐵)
20	19	exlimivv 1935	. 2 ⊢ (∃𝑓∃𝑔(𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐴 ≈ 𝐵)
21	7, 20	sylbi 216	1 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2709  Vcvv 3474   ∖ cdif 3944   ∪ cun 3945   ⊆ wss 3947  ∪ cuni 4907   class class class wbr 5147  ^◡ccnv 5674   ↾ cres 5677   “ cima 5678  –1-1→wf1 6537  –1-1-onto→wf1o 6539   ≈ cen 8932   ≼ cdom 8933

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-pow 5362  ax-pr 5426  ax-un 7721

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-pw 4603  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  df-en 8936  df-dom 8937

This theorem is referenced by:  sbth  9089