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Mirrors > Home > MPE Home > Th. List > Mathboxes > elrnmpt1d | Structured version Visualization version GIF version |
Description: Elementhood in an image set. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
elrnmpt1d.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
elrnmpt1d.2 | ⊢ (𝜑 → 𝑥 ∈ 𝐴) |
elrnmpt1d.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
Ref | Expression |
---|---|
elrnmpt1d | ⊢ (𝜑 → 𝐵 ∈ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrnmpt1d.2 | . 2 ⊢ (𝜑 → 𝑥 ∈ 𝐴) | |
2 | elrnmpt1d.3 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
3 | elrnmpt1d.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | elrnmpt1 5829 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ ran 𝐹) |
5 | 1, 2, 4 | syl2anc 586 | 1 ⊢ (𝜑 → 𝐵 ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ↦ cmpt 5145 ran crn 5555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-mpt 5146 df-cnv 5562 df-dm 5564 df-rn 5565 |
This theorem is referenced by: rnmptbd2lem 41518 rnmptbdlem 41525 rnmptss2 41527 rnmptssbi 41532 supminfxrrnmpt 41745 |
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