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Mirrors > Home > MPE Home > Th. List > f1dom4g | Structured version Visualization version GIF version |
Description: The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg 8967 does not require the Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.) |
Ref | Expression |
---|---|
f1dom4g | ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1eq1 6775 | . . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1→𝐵)) | |
2 | 1 | spcegv 3581 | . . . 4 ⊢ (𝐹 ∈ 𝑉 → (𝐹:𝐴–1-1→𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
3 | 2 | imp 406 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → ∃𝑓 𝑓:𝐴–1-1→𝐵) |
4 | 3 | 3ad2antl1 1182 | . 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴–1-1→𝐵) → ∃𝑓 𝑓:𝐴–1-1→𝐵) |
5 | brdom2g 8950 | . . . 4 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) | |
6 | 5 | 3adant1 1127 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
7 | 6 | adantr 480 | . 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴–1-1→𝐵) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
8 | 4, 7 | mpbird 257 | 1 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 ∃wex 1773 ∈ wcel 2098 class class class wbr 5141 –1-1→wf1 6533 ≼ cdom 8936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-dom 8940 |
This theorem is referenced by: domssl 8993 domssr 8994 |
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