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Mirrors > Home > MPE Home > Th. List > brdomg | Structured version Visualization version GIF version |
Description: Dominance relation. (Contributed by NM, 15-Jun-1998.) |
Ref | Expression |
---|---|
brdomg | ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1eq2 6571 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝑦)) | |
2 | 1 | exbidv 1922 | . . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝑦)) |
3 | f1eq3 6572 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑓:𝐴–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝐵)) | |
4 | 3 | exbidv 1922 | . . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
5 | df-dom 8511 | . . . 4 ⊢ ≼ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} | |
6 | 2, 4, 5 | brabg 5426 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐶) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
7 | 6 | ex 415 | . 2 ⊢ (𝐴 ∈ V → (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))) |
8 | reldom 8515 | . . . . 5 ⊢ Rel ≼ | |
9 | 8 | brrelex1i 5608 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V) |
10 | f1f 6575 | . . . . . 6 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵) | |
11 | fdm 6522 | . . . . . . 7 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴) | |
12 | vex 3497 | . . . . . . . 8 ⊢ 𝑓 ∈ V | |
13 | 12 | dmex 7616 | . . . . . . 7 ⊢ dom 𝑓 ∈ V |
14 | 11, 13 | eqeltrrdi 2922 | . . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V) |
15 | 10, 14 | syl 17 | . . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V) |
16 | 15 | exlimiv 1931 | . . . 4 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V) |
17 | 9, 16 | pm5.21ni 381 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
18 | 17 | a1d 25 | . 2 ⊢ (¬ 𝐴 ∈ V → (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))) |
19 | 7, 18 | pm2.61i 184 | 1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 = wceq 1537 ∃wex 1780 ∈ wcel 2114 Vcvv 3494 class class class wbr 5066 dom cdm 5555 ⟶wf 6351 –1-1→wf1 6352 ≼ cdom 8507 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 df-fn 6358 df-f 6359 df-f1 6360 df-dom 8511 |
This theorem is referenced by: brdomi 8520 brdom 8521 f1dom2g 8527 f1domg 8529 dom3d 8551 domdifsn 8600 fidomtri 9422 hashdom 13741 hashge3el3dif 13845 sizusglecusg 27245 erdsze2lem1 32450 |
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