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Mirrors > Home > ILE Home > Th. List > sbthlemi10 | GIF version |
Description: Lemma for isbth 6944. (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 |
---|---|
sbthlemi10 | ⊢ ((EXMID ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) → 𝐴 ≈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbthlem.4 | . . . . . 6 ⊢ 𝐵 ∈ V | |
2 | 1 | brdom 6728 | . . . . 5 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵) |
3 | sbthlem.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
4 | 3 | brdom 6728 | . . . . 5 ⊢ (𝐵 ≼ 𝐴 ↔ ∃𝑔 𝑔:𝐵–1-1→𝐴) |
5 | 2, 4 | anbi12i 457 | . . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) ↔ (∃𝑓 𝑓:𝐴–1-1→𝐵 ∧ ∃𝑔 𝑔:𝐵–1-1→𝐴)) |
6 | eeanv 1925 | . . . 4 ⊢ (∃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 186 | . . 3 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) ↔ ∃𝑓∃𝑔(𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴)) |
8 | sbthlem.3 | . . . . . . 7 ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) | |
9 | vex 2733 | . . . . . . . . 9 ⊢ 𝑓 ∈ V | |
10 | 9 | resex 4932 | . . . . . . . 8 ⊢ (𝑓 ↾ ∪ 𝐷) ∈ V |
11 | vex 2733 | . . . . . . . . . 10 ⊢ 𝑔 ∈ V | |
12 | 11 | cnvex 5149 | . . . . . . . . 9 ⊢ ◡𝑔 ∈ V |
13 | 12 | resex 4932 | . . . . . . . 8 ⊢ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) ∈ V |
14 | 10, 13 | unex 4426 | . . . . . . 7 ⊢ ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ∈ V |
15 | 8, 14 | eqeltri 2243 | . . . . . 6 ⊢ 𝐻 ∈ V |
16 | sbthlem.2 | . . . . . . 7 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} | |
17 | 3, 16, 8 | sbthlemi9 6942 | . . . . . 6 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐻:𝐴–1-1-onto→𝐵) |
18 | f1oen3g 6732 | . . . . . 6 ⊢ ((𝐻 ∈ V ∧ 𝐻:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | |
19 | 15, 17, 18 | sylancr 412 | . . . . 5 ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐴 ≈ 𝐵) |
20 | 19 | 3expib 1201 | . . . 4 ⊢ (EXMID → ((𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐴 ≈ 𝐵)) |
21 | 20 | exlimdvv 1890 | . . 3 ⊢ (EXMID → (∃𝑓∃𝑔(𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐴 ≈ 𝐵)) |
22 | 7, 21 | syl5bi 151 | . 2 ⊢ (EXMID → ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)) |
23 | 22 | imp 123 | 1 ⊢ ((EXMID ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) → 𝐴 ≈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 973 = wceq 1348 ∃wex 1485 ∈ wcel 2141 {cab 2156 Vcvv 2730 ∖ cdif 3118 ∪ cun 3119 ⊆ wss 3121 ∪ cuni 3796 class class class wbr 3989 EXMIDwem 4180 ◡ccnv 4610 ↾ cres 4613 “ cima 4614 –1-1→wf1 5195 –1-1-onto→wf1o 5197 ≈ cen 6716 ≼ cdom 6717 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-exmid 4181 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-en 6719 df-dom 6720 |
This theorem is referenced by: isbth 6944 |
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