![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dom2 | 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. 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 2724 | . 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 8984 | . 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 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 class class class wbr 5138 ≼ cdom 8932 |
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-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 |
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-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-dom 8936 |
This theorem is referenced by: infpwfidom 10018 rpnnen1lem6 12962 rpnnen2lem12 16164 tgdom 22802 vitali 25463 rpnnen3 42226 |
Copyright terms: Public domain | W3C validator |