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| Description: Elementhood in an image set. Deducion version of elrnmpt1 5970. (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 5970 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ ran 𝐹) | 
| 5 | 1, 2, 4 | syl2anc 584 | 1 ⊢ (𝜑 → 𝐵 ∈ ran 𝐹) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ↦ cmpt 5224 ran crn 5685 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-mpt 5225 df-cnv 5692 df-dm 5694 df-rn 5695 | 
| This theorem is referenced by: mptcnfimad 8012 elrgspnsubrunlem1 33252 rnmptbd2lem 45260 rnmptbdlem 45267 rnmptss2 45269 rnmptssbi 45272 supminfxrrnmpt 45487 sge0f1o 46402 | 
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