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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 2610 | . . 3 ⊢ 𝐶 = 𝐶 | |
2 | elrnmpt1s.1 | . . . . 5 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) | |
3 | 2 | eqeq2d 2620 | . . . 4 ⊢ (𝑥 = 𝐷 → (𝐶 = 𝐵 ↔ 𝐶 = 𝐶)) |
4 | 3 | rspcev 3282 | . . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 = 𝐶) → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) |
5 | 1, 4 | mpan2 703 | . 2 ⊢ (𝐷 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) |
6 | rnmpt.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
7 | 6 | elrnmpt 5293 | . . 3 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) |
8 | 7 | biimparc 503 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝐶 = 𝐵 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹) |
9 | 5, 8 | sylan 487 | 1 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ↦ cmpt 4643 ran crn 5039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-mpt 4645 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: wunex2 9439 dfod2 17804 dprd2dlem1 18263 dprd2da 18264 ordtbaslem 20802 subgntr 21720 opnsubg 21721 tgpconcomp 21726 tsmsxplem1 21766 xrge0gsumle 22444 xrge0tsms 22445 minveclem3b 23007 minveclem3 23008 minveclem4 23011 efsubm 24101 dchrisum0fno1 25000 xrge0tsmsd 29116 esumcvg 29475 esum2d 29482 msubco 30682 sge0xaddlem1 39326 |
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