Theorem sbthlem10 9153

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)))
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 9025	. . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴⟶1-1𝐵)
3		sbthlem.1	. . . . 5 ⊢ 𝐴 ∈ V
4	3	brdom 9025	. . . 4 ⊢ (𝐵 ≼ 𝐴 ↔ ∃𝑔 𝑔:𝐵⟶1-1𝐴)
5	2, 4	anbi12i 640	. . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) ↔ (∃𝑓 𝑓:𝐴⟶1-1𝐵 ∧ ∃𝑔 𝑔:𝐵⟶1-1𝐴))
6		exdistrv 1990	. . 3 ⊢ (∃𝑓∃𝑔(𝑓:𝐴⟶1-1𝐵 ∧ 𝑔:𝐵⟶1-1𝐴) ↔ (∃𝑓 𝑓:𝐴⟶1-1𝐵 ∧ ∃𝑔 𝑔:𝐵⟶1-1𝐴))
7	5, 6	bitr4i 281	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) ↔ ∃𝑓∃𝑔(𝑓:𝐴⟶1-1𝐵 ∧ 𝑔:𝐵⟶1-1𝐴))
8		sbthlem.3	. . . . 5 ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
9		vex 3458	. . . . . . 7 ⊢ 𝑓 ∈ V
10	9	resex 6036	. . . . . 6 ⊢ (𝑓 ↾ ∪ 𝐷) ∈ V
11		vex 3458	. . . . . . . 8 ⊢ 𝑔 ∈ V
12	11	cnvex 7971	. . . . . . 7 ⊢ ◡𝑔 ∈ V
13	12	resex 6036	. . . . . 6 ⊢ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) ∈ V
14	10, 13	unex 7793	. . . . 5 ⊢ ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ∈ V
15	8, 14	eqeltri 2860	. . . 4 ⊢ 𝐻 ∈ V
16		sbthlem.2	. . . . 5 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}
17	3, 16, 8	sbthlem9 9152	. . . 4 ⊢ ((𝑓:𝐴⟶1-1𝐵 ∧ 𝑔:𝐵⟶1-1𝐴) → 𝐻:𝐴⟶1-1onto𝐵)
18		f1oen3g 9031	. . . 4 ⊢ ((𝐻 ∈ V ∧ 𝐻:𝐴⟶1-1onto𝐵) → 𝐴 ≈ 𝐵)
19	15, 17, 18	sylancr 599	. . 3 ⊢ ((𝑓:𝐴⟶1-1𝐵 ∧ 𝑔:𝐵⟶1-1𝐴) → 𝐴 ≈ 𝐵)
20	19	exlimivv 1967	. 2 ⊢ (∃𝑓∃𝑔(𝑓:𝐴⟶1-1𝐵 ∧ 𝑔:𝐵⟶1-1𝐴) → 𝐴 ≈ 𝐵)
21	7, 20	sylbi 220	1 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   ∪ cun 4152  ∪ cuni 4872   class class class wbr 5108  ^◡ccnv 5657   ↾ cres 5660   “ cima 5661  ⟶1-1wf1 6563  ⟶1-1ontowf1o 6565   ≈ cen 9008   ≼ cdom 9009

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-pow 5334  ax-pr 5398  ax-un 7783

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-pw 4562  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  df-en 9012  df-dom 9013

This theorem is used by:  sbth  9154