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Mirrors > Home > MPE Home > Th. List > sbthlem10 | Structured version Visualization version GIF version |
Description: Lemma for sbth 8640. (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 8524 | . . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵) |
3 | sbthlem.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
4 | 3 | brdom 8524 | . . . 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 280 | . 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 3500 | . . . . . . 7 ⊢ 𝑓 ∈ V | |
10 | 9 | resex 5902 | . . . . . 6 ⊢ (𝑓 ↾ ∪ 𝐷) ∈ V |
11 | vex 3500 | . . . . . . . 8 ⊢ 𝑔 ∈ V | |
12 | 11 | cnvex 7633 | . . . . . . 7 ⊢ ◡𝑔 ∈ V |
13 | 12 | resex 5902 | . . . . . 6 ⊢ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) ∈ V |
14 | 10, 13 | unex 7472 | . . . . 5 ⊢ ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ∈ V |
15 | 8, 14 | eqeltri 2912 | . . . 4 ⊢ 𝐻 ∈ V |
16 | sbthlem.2 | . . . . 5 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} | |
17 | 3, 16, 8 | sbthlem9 8638 | . . . 4 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐻:𝐴–1-1-onto→𝐵) |
18 | f1oen3g 8528 | . . . 4 ⊢ ((𝐻 ∈ V ∧ 𝐻:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | |
19 | 15, 17, 18 | sylancr 589 | . . 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 219 | 1 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∃wex 1779 ∈ wcel 2113 {cab 2802 Vcvv 3497 ∖ cdif 3936 ∪ cun 3937 ⊆ wss 3939 ∪ cuni 4841 class class class wbr 5069 ◡ccnv 5557 ↾ cres 5560 “ cima 5561 –1-1→wf1 6355 –1-1-onto→wf1o 6357 ≈ cen 8509 ≼ cdom 8510 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-en 8513 df-dom 8514 |
This theorem is referenced by: sbth 8640 |
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