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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 5830 | . 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 1537 ∈ wcel 2114 ↦ cmpt 5146 ran crn 5556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-mpt 5147 df-cnv 5563 df-dm 5565 df-rn 5566 |
This theorem is referenced by: rnmptbd2lem 41540 rnmptbdlem 41547 rnmptss2 41549 rnmptssbi 41554 supminfxrrnmpt 41767 |
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