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Mirrors > Home > MPE Home > Th. List > domen | Structured version Visualization version GIF version |
Description: Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146. (Contributed by NM, 15-Jun-1998.) |
Ref | Expression |
---|---|
bren.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
domen | ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bren.1 | . . 3 ⊢ 𝐵 ∈ V | |
2 | 1 | brdom 8523 | . 2 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵) |
3 | vex 3499 | . . . . . 6 ⊢ 𝑓 ∈ V | |
4 | 3 | f11o 7650 | . . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 ↔ ∃𝑥(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
5 | 4 | exbii 1848 | . . . 4 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 ↔ ∃𝑓∃𝑥(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
6 | excom 2169 | . . . 4 ⊢ (∃𝑓∃𝑥(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑥∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) | |
7 | 5, 6 | bitri 277 | . . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 ↔ ∃𝑥∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
8 | bren 8520 | . . . . . 6 ⊢ (𝐴 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑥) | |
9 | 8 | anbi1i 625 | . . . . 5 ⊢ ((𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (∃𝑓 𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
10 | 19.41v 1950 | . . . . 5 ⊢ (∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (∃𝑓 𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) | |
11 | 9, 10 | bitr4i 280 | . . . 4 ⊢ ((𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
12 | 11 | exbii 1848 | . . 3 ⊢ (∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑥∃𝑓(𝑓:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
13 | 7, 12 | bitr4i 280 | . 2 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) |
14 | 2, 13 | bitri 277 | 1 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∃wex 1780 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 class class class wbr 5068 –1-1→wf1 6354 –1-1-onto→wf1o 6356 ≈ cen 8508 ≼ cdom 8509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 df-dm 5567 df-rn 5568 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-en 8512 df-dom 8513 |
This theorem is referenced by: domeng 8525 infcntss 8794 ramub2 16352 ram0 16360 |
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