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Mirrors > Home > MPE Home > Th. List > brdom2g | Structured version Visualization version GIF version |
Description: Dominance relation. This variation of brdomg 8854 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of brdomg 8854. (Revised by BTernaryTau, 29-Nov-2024.) |
Ref | Expression |
---|---|
brdom2g | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1eq2 6731 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝑦)) | |
2 | 1 | exbidv 1924 | . 2 ⊢ (𝑥 = 𝐴 → (∃𝑓 𝑓:𝑥–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝑦)) |
3 | f1eq3 6732 | . . 3 ⊢ (𝑦 = 𝐵 → (𝑓:𝐴–1-1→𝑦 ↔ 𝑓:𝐴–1-1→𝐵)) | |
4 | 3 | exbidv 1924 | . 2 ⊢ (𝑦 = 𝐵 → (∃𝑓 𝑓:𝐴–1-1→𝑦 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
5 | df-dom 8843 | . 2 ⊢ ≼ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} | |
6 | 2, 4, 5 | brabg 5494 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 class class class wbr 5103 –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 |
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-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-br 5104 df-opab 5166 df-fn 6496 df-f 6497 df-f1 6498 df-dom 8843 |
This theorem is referenced by: brdomg 8854 brdomi 8856 f1dom4g 8863 0domg 9002 |
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