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| Mirrors > Home > MPE Home > Th. List > Mathboxes > elrnmpt1sf | Structured version Visualization version GIF version | ||
| Description: Elementhood in an image set. Same as elrnmpt1s 5970, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| Ref | Expression |
|---|---|
| elrnmpt1sf.1 | ⊢ Ⅎ𝑥𝐶 |
| elrnmpt1sf.2 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
| elrnmpt1sf.3 | ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) |
| Ref | Expression |
|---|---|
| elrnmpt1sf | ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ 𝐶 = 𝐶 | |
| 2 | elrnmpt1sf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
| 3 | 2, 2 | nfeq 2919 | . . . 4 ⊢ Ⅎ𝑥 𝐶 = 𝐶 |
| 4 | elrnmpt1sf.3 | . . . . 5 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) | |
| 5 | 4 | eqeq2d 2748 | . . . 4 ⊢ (𝑥 = 𝐷 → (𝐶 = 𝐵 ↔ 𝐶 = 𝐶)) |
| 6 | 3, 5 | rspce 3611 | . . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 = 𝐶) → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) |
| 7 | 1, 6 | mpan2 691 | . 2 ⊢ (𝐷 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) |
| 8 | elrnmpt1sf.2 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 9 | 2, 8 | elrnmptf 45186 | . . 3 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) |
| 10 | 9 | biimparc 479 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝐶 = 𝐵 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹) |
| 11 | 7, 10 | sylan 580 | 1 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 ∃wrex 3070 ↦ cmpt 5225 ran crn 5686 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-cnv 5693 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: sge0f1o 46397 |
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