| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > sbthlem10 | Structured version Visualization version GIF version | ||
| Description: Lemma for sbth 9133. (Contributed by NM, 28-Mar-1998.) |
| 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 |
| Ref | Expression |
|---|---|
| sbthlem10 | ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbthlem.4 | . . . . 5 ⊢ 𝐵 ∈ V | |
| 2 | 1 | brdom 9001 | . . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵) |
| 3 | sbthlem.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
| 4 | 3 | brdom 9001 | . . . 4 ⊢ (𝐵 ≼ 𝐴 ↔ ∃𝑔 𝑔:𝐵–1-1→𝐴) |
| 5 | 2, 4 | anbi12i 628 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) ↔ (∃𝑓 𝑓:𝐴–1-1→𝐵 ∧ ∃𝑔 𝑔:𝐵–1-1→𝐴)) |
| 6 | exdistrv 1955 | . . 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 278 | . 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 3484 | . . . . . . 7 ⊢ 𝑓 ∈ V | |
| 10 | 9 | resex 6047 | . . . . . 6 ⊢ (𝑓 ↾ ∪ 𝐷) ∈ V |
| 11 | vex 3484 | . . . . . . . 8 ⊢ 𝑔 ∈ V | |
| 12 | 11 | cnvex 7947 | . . . . . . 7 ⊢ ◡𝑔 ∈ V |
| 13 | 12 | resex 6047 | . . . . . 6 ⊢ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) ∈ V |
| 14 | 10, 13 | unex 7764 | . . . . 5 ⊢ ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ∈ V |
| 15 | 8, 14 | eqeltri 2837 | . . . 4 ⊢ 𝐻 ∈ V |
| 16 | sbthlem.2 | . . . . 5 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} | |
| 17 | 3, 16, 8 | sbthlem9 9131 | . . . 4 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐻:𝐴–1-1-onto→𝐵) |
| 18 | f1oen3g 9007 | . . . 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 1932 | . 2 ⊢ (∃𝑓∃𝑔(𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐴 ≈ 𝐵) |
| 21 | 7, 20 | sylbi 217 | 1 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ⊆ wss 3951 ∪ cuni 4907 class class class wbr 5143 ◡ccnv 5684 ↾ cres 5687 “ cima 5688 –1-1→wf1 6558 –1-1-onto→wf1o 6560 ≈ cen 8982 ≼ cdom 8983 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-en 8986 df-dom 8987 |
| This theorem is referenced by: sbth 9133 |
| Copyright terms: Public domain | W3C validator |