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Mirrors > Home > MPE Home > Th. List > sbthlem10 | Structured version Visualization version GIF version |
Description: Lemma for sbth 9089. (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 8952 | . . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵) |
3 | sbthlem.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
4 | 3 | brdom 8952 | . . . 4 ⊢ (𝐵 ≼ 𝐴 ↔ ∃𝑔 𝑔:𝐵–1-1→𝐴) |
5 | 2, 4 | anbi12i 626 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) ↔ (∃𝑓 𝑓:𝐴–1-1→𝐵 ∧ ∃𝑔 𝑔:𝐵–1-1→𝐴)) |
6 | exdistrv 1951 | . . 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 3470 | . . . . . . 7 ⊢ 𝑓 ∈ V | |
10 | 9 | resex 6019 | . . . . . 6 ⊢ (𝑓 ↾ ∪ 𝐷) ∈ V |
11 | vex 3470 | . . . . . . . 8 ⊢ 𝑔 ∈ V | |
12 | 11 | cnvex 7909 | . . . . . . 7 ⊢ ◡𝑔 ∈ V |
13 | 12 | resex 6019 | . . . . . 6 ⊢ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) ∈ V |
14 | 10, 13 | unex 7726 | . . . . 5 ⊢ ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ∈ V |
15 | 8, 14 | eqeltri 2821 | . . . 4 ⊢ 𝐻 ∈ V |
16 | sbthlem.2 | . . . . 5 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} | |
17 | 3, 16, 8 | sbthlem9 9087 | . . . 4 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐻:𝐴–1-1-onto→𝐵) |
18 | f1oen3g 8958 | . . . 4 ⊢ ((𝐻 ∈ V ∧ 𝐻:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | |
19 | 15, 17, 18 | sylancr 586 | . . 3 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐴 ≈ 𝐵) |
20 | 19 | exlimivv 1927 | . 2 ⊢ (∃𝑓∃𝑔(𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐴 ≈ 𝐵) |
21 | 7, 20 | sylbi 216 | 1 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∃wex 1773 ∈ wcel 2098 {cab 2701 Vcvv 3466 ∖ cdif 3937 ∪ cun 3938 ⊆ wss 3940 ∪ cuni 4899 class class class wbr 5138 ◡ccnv 5665 ↾ cres 5668 “ cima 5669 –1-1→wf1 6530 –1-1-onto→wf1o 6532 ≈ cen 8932 ≼ cdom 8933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-en 8936 df-dom 8937 |
This theorem is referenced by: sbth 9089 |
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