![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > elrnmpt1s | Structured version Visualization version GIF version |
Description: Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.) |
Ref | Expression |
---|---|
rnmpt.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
elrnmpt1s.1 | ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
elrnmpt1s | ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . 3 ⊢ 𝐶 = 𝐶 | |
2 | elrnmpt1s.1 | . . . 4 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) | |
3 | 2 | rspceeqv 3628 | . . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 = 𝐶) → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) |
4 | 1, 3 | mpan2 689 | . 2 ⊢ (𝐷 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) |
5 | rnmpt.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
6 | 5 | elrnmpt 5958 | . . 3 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) |
7 | 6 | biimparc 478 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝐶 = 𝐵 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹) |
8 | 4, 7 | sylan 578 | 1 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3059 ↦ cmpt 5232 ran crn 5679 |
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 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-mpt 5233 df-cnv 5686 df-dm 5688 df-rn 5689 |
This theorem is referenced by: wunex2 10763 dfod2 19531 dprd2dlem1 20010 dprd2da 20011 ordtbaslem 23136 subgntr 24055 opnsubg 24056 tgpconncomp 24061 tsmsxplem1 24101 xrge0gsumle 24793 xrge0tsms 24794 minveclem3b 25400 minveclem3 25401 minveclem4 25404 efsubm 26530 dchrisum0fno1 27489 fnpreimac 32538 xrge0tsmsd 32861 esumcvg 33836 esum2d 33843 msubco 35272 suprubrnmpt2 44766 infxrlbrnmpt2 44930 sge0xaddlem1 45959 |
Copyright terms: Public domain | W3C validator |