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Mirrors > Home > MPE Home > Th. List > dom3d | Structured version Visualization version GIF version |
Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by Mario Carneiro, 20-May-2013.) |
Ref | Expression |
---|---|
dom2d.1 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)) |
dom2d.2 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦))) |
dom3d.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
dom3d.4 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
Ref | Expression |
---|---|
dom3d | ⊢ (𝜑 → 𝐴 ≼ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dom2d.1 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)) | |
2 | dom2d.2 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦))) | |
3 | 1, 2 | dom2lem 8552 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵) |
4 | f1f 6578 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵) |
6 | dom3d.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
7 | dom3d.4 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
8 | fex2 7641 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ V) | |
9 | 5, 6, 7, 8 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ V) |
10 | f1eq1 6573 | . . 3 ⊢ (𝑧 = (𝑥 ∈ 𝐴 ↦ 𝐶) → (𝑧:𝐴–1-1→𝐵 ↔ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵)) | |
11 | 9, 3, 10 | spcedv 3602 | . 2 ⊢ (𝜑 → ∃𝑧 𝑧:𝐴–1-1→𝐵) |
12 | brdomg 8522 | . . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ≼ 𝐵 ↔ ∃𝑧 𝑧:𝐴–1-1→𝐵)) | |
13 | 7, 12 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 ≼ 𝐵 ↔ ∃𝑧 𝑧:𝐴–1-1→𝐵)) |
14 | 11, 13 | mpbird 259 | 1 ⊢ (𝜑 → 𝐴 ≼ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∃wex 1779 ∈ wcel 2113 Vcvv 3497 class class class wbr 5069 ↦ cmpt 5149 ⟶wf 6354 –1-1→wf1 6355 ≼ cdom 8510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fv 6366 df-dom 8514 |
This theorem is referenced by: dom3 8556 xpdom2 8615 fopwdom 8628 |
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