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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 5373, 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 2622 | . . 3 ⊢ 𝐶 = 𝐶 | |
2 | 1 | nfth 1727 | . . . 4 ⊢ Ⅎ𝑥 𝐶 = 𝐶 |
3 | elrnmpt1sf.3 | . . . . 5 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) | |
4 | 3 | eqeq2d 2632 | . . . 4 ⊢ (𝑥 = 𝐷 → (𝐶 = 𝐵 ↔ 𝐶 = 𝐶)) |
5 | 2, 4 | rspce 3304 | . . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 = 𝐶) → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) |
6 | 1, 5 | mpan2 707 | . 2 ⊢ (𝐷 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) |
7 | elrnmpt1sf.1 | . . . 4 ⊢ Ⅎ𝑥𝐶 | |
8 | elrnmpt1sf.2 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
9 | 7, 8 | elrnmptf 39367 | . . 3 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) |
10 | 9 | biimparc 504 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝐶 = 𝐵 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹) |
11 | 6, 10 | sylan 488 | 1 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Ⅎwnfc 2751 ∃wrex 2913 ↦ cmpt 4729 ran crn 5115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-mpt 4730 df-cnv 5122 df-dm 5124 df-rn 5125 |
This theorem is referenced by: sge0f1o 40599 |
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