| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > sbthlem10 | Structured version Visualization version GIF version | ||
| Description: Lemma for sbth 9092. (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 8963 | . . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵) |
| 3 | sbthlem.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
| 4 | 3 | brdom 8963 | . . . 4 ⊢ (𝐵 ≼ 𝐴 ↔ ∃𝑔 𝑔:𝐵–1-1→𝐴) |
| 5 | 2, 4 | anbi12i 640 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) ↔ (∃𝑓 𝑓:𝐴–1-1→𝐵 ∧ ∃𝑔 𝑔:𝐵–1-1→𝐴)) |
| 6 | exdistrv 1988 | . . 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 281 | . 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 3461 | . . . . . . 7 ⊢ 𝑓 ∈ V | |
| 10 | 9 | resex 6030 | . . . . . 6 ⊢ (𝑓 ↾ ∪ 𝐷) ∈ V |
| 11 | vex 3461 | . . . . . . . 8 ⊢ 𝑔 ∈ V | |
| 12 | 11 | cnvex 7928 | . . . . . . 7 ⊢ ◡𝑔 ∈ V |
| 13 | 12 | resex 6030 | . . . . . 6 ⊢ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) ∈ V |
| 14 | 10, 13 | unex 7752 | . . . . 5 ⊢ ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ∈ V |
| 15 | 8, 14 | eqeltri 2861 | . . . 4 ⊢ 𝐻 ∈ V |
| 16 | sbthlem.2 | . . . . 5 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} | |
| 17 | 3, 16, 8 | sbthlem9 9090 | . . . 4 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐻:𝐴–1-1-onto→𝐵) |
| 18 | f1oen3g 8969 | . . . 4 ⊢ ((𝐻 ∈ V ∧ 𝐻:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | |
| 19 | 15, 17, 18 | sylancr 599 | . . 3 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐴 ≈ 𝐵) |
| 20 | 19 | exlimivv 1965 | . 2 ⊢ (∃𝑓∃𝑔(𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐴 ≈ 𝐵) |
| 21 | 7, 20 | sylbi 220 | 1 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∃wex 1812 ∈ wcel 2146 {cab 2743 Vcvv 3457 ∖ cdif 3903 ∪ cun 3904 ⊆ wss 3906 ∪ cuni 4874 class class class wbr 5111 ◡ccnv 5662 ↾ cres 5665 “ cima 5666 –1-1→wf1 6537 –1-1-onto→wf1o 6539 ≈ cen 8946 ≼ cdom 8947 |
| 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 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-pow 5338 ax-pr 5406 ax-un 7742 |
| 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 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-br 5112 df-opab 5176 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-en 8950 df-dom 8951 |
| This theorem is used by: sbth 9092 |
| Copyright terms: Public domain | W3C validator |