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| Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. 𝐶 and 𝐷 can be read 𝐶(𝑥) and 𝐷(𝑦), as can be inferred from their distinct variable conditions. (Contributed by NM, 26-Oct-2003.) | 
| Ref | Expression | 
|---|---|
| dom2.1 | ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵) | 
| dom2.2 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦)) | 
| Ref | Expression | 
|---|---|
| dom2 | ⊢ (𝐵 ∈ 𝑉 → 𝐴 ≼ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2737 | . 2 ⊢ 𝐴 = 𝐴 | |
| 2 | dom2.1 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵) | |
| 3 | 2 | a1i 11 | . . 3 ⊢ (𝐴 = 𝐴 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)) | 
| 4 | dom2.2 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦)) | |
| 5 | 4 | a1i 11 | . . 3 ⊢ (𝐴 = 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦))) | 
| 6 | 3, 5 | dom2d 9033 | . 2 ⊢ (𝐴 = 𝐴 → (𝐵 ∈ 𝑉 → 𝐴 ≼ 𝐵)) | 
| 7 | 1, 6 | ax-mp 5 | 1 ⊢ (𝐵 ∈ 𝑉 → 𝐴 ≼ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ≼ cdom 8983 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-dom 8987 | 
| This theorem is referenced by: infpwfidom 10068 rpnnen1lem6 13024 rpnnen2lem12 16261 tgdom 22985 vitali 25648 rpnnen3 43044 | 
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