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Mirrors > Home > MPE Home > Th. List > brdomgOLD | Structured version Visualization version GIF version |
Description: Obsolete version of brdomg 8854 as of 29-Nov-2024. (Contributed by NM, 15-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
brdomgOLD | ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1eq2 6731 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝑦)) | |
2 | 1 | exbidv 1924 | . . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝑦)) |
3 | f1eq3 6732 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑓:𝐴–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝐵)) | |
4 | 3 | exbidv 1924 | . . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
5 | df-dom 8843 | . . . 4 ⊢ ≼ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} | |
6 | 2, 4, 5 | brabg 5494 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐶) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
7 | 6 | ex 413 | . 2 ⊢ (𝐴 ∈ V → (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))) |
8 | reldom 8847 | . . . . 5 ⊢ Rel ≼ | |
9 | 8 | brrelex1i 5686 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V) |
10 | f1f 6735 | . . . . . 6 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵) | |
11 | fdm 6674 | . . . . . . 7 ⊢ (𝑓:𝐴⟶𝐵 → dom 𝑓 = 𝐴) | |
12 | vex 3447 | . . . . . . . 8 ⊢ 𝑓 ∈ V | |
13 | 12 | dmex 7840 | . . . . . . 7 ⊢ dom 𝑓 ∈ V |
14 | 11, 13 | eqeltrrdi 2847 | . . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝐴 ∈ V) |
15 | 10, 14 | syl 17 | . . . . 5 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V) |
16 | 15 | exlimiv 1933 | . . . 4 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 → 𝐴 ∈ V) |
17 | 9, 16 | pm5.21ni 378 | . . 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 182 | 1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1541 ∃wex 1781 ∈ wcel 2106 Vcvv 3443 class class class wbr 5103 dom cdm 5631 ⟶wf 6489 –1-1→wf1 6490 ≼ cdom 8839 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-xp 5637 df-rel 5638 df-cnv 5639 df-dm 5641 df-rn 5642 df-fn 6496 df-f 6497 df-f1 6498 df-dom 8843 |
This theorem is referenced by: (None) |
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