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Mirrors > Home > MPE Home > Th. List > elrnmpt1d | Structured version Visualization version GIF version |
Description: Elementhood in an image set. Deducion version of elrnmpt1 5974. (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 5974 | . 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 1537 ∈ wcel 2106 ↦ cmpt 5231 ran crn 5690 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-mpt 5232 df-cnv 5697 df-dm 5699 df-rn 5700 |
This theorem is referenced by: mptcnfimad 8010 rnmptbd2lem 45193 rnmptbdlem 45200 rnmptss2 45202 rnmptssbi 45206 supminfxrrnmpt 45421 sge0f1o 46338 |
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