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Mirrors > Home > MPE Home > Th. List > sbthlem10 | Structured version Visualization version GIF version |
Description: Lemma for sbth 8621. (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 8504 | . . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵) |
3 | sbthlem.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
4 | 3 | brdom 8504 | . . . 4 ⊢ (𝐵 ≼ 𝐴 ↔ ∃𝑔 𝑔:𝐵–1-1→𝐴) |
5 | 2, 4 | anbi12i 629 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) ↔ (∃𝑓 𝑓:𝐴–1-1→𝐵 ∧ ∃𝑔 𝑔:𝐵–1-1→𝐴)) |
6 | exdistrv 1956 | . . 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 3444 | . . . . . . 7 ⊢ 𝑓 ∈ V | |
10 | 9 | resex 5866 | . . . . . 6 ⊢ (𝑓 ↾ ∪ 𝐷) ∈ V |
11 | vex 3444 | . . . . . . . 8 ⊢ 𝑔 ∈ V | |
12 | 11 | cnvex 7612 | . . . . . . 7 ⊢ ◡𝑔 ∈ V |
13 | 12 | resex 5866 | . . . . . 6 ⊢ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) ∈ V |
14 | 10, 13 | unex 7449 | . . . . 5 ⊢ ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ∈ V |
15 | 8, 14 | eqeltri 2886 | . . . 4 ⊢ 𝐻 ∈ V |
16 | sbthlem.2 | . . . . 5 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} | |
17 | 3, 16, 8 | sbthlem9 8619 | . . . 4 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐻:𝐴–1-1-onto→𝐵) |
18 | f1oen3g 8508 | . . . 4 ⊢ ((𝐻 ∈ V ∧ 𝐻:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | |
19 | 15, 17, 18 | sylancr 590 | . . 3 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐴 ≈ 𝐵) |
20 | 19 | exlimivv 1933 | . 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 |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 {cab 2776 Vcvv 3441 ∖ cdif 3878 ∪ cun 3879 ⊆ wss 3881 ∪ cuni 4800 class class class wbr 5030 ◡ccnv 5518 ↾ cres 5521 “ cima 5522 –1-1→wf1 6321 –1-1-onto→wf1o 6323 ≈ cen 8489 ≼ cdom 8490 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-en 8493 df-dom 8494 |
This theorem is referenced by: sbth 8621 |
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