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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 5294, 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 2610 | . . 3 ⊢ 𝐶 = 𝐶 | |
2 | 1 | nfth 1718 | . . . 4 ⊢ Ⅎ𝑥 𝐶 = 𝐶 |
3 | elrnmpt1sf.3 | . . . . 5 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) | |
4 | 3 | eqeq2d 2620 | . . . 4 ⊢ (𝑥 = 𝐷 → (𝐶 = 𝐵 ↔ 𝐶 = 𝐶)) |
5 | 2, 4 | rspce 3277 | . . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 = 𝐶) → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) |
6 | 1, 5 | mpan2 703 | . 2 ⊢ (𝐷 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) |
7 | elrnmpt1sf.1 | . . . 4 ⊢ Ⅎ𝑥𝐶 | |
8 | elrnmpt1sf.2 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
9 | 7, 8 | elrnmptf 38362 | . . 3 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) |
10 | 9 | biimparc 503 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝐶 = 𝐵 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹) |
11 | 6, 10 | sylan 487 | 1 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Ⅎwnfc 2738 ∃wrex 2897 ↦ cmpt 4643 ran crn 5039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-mpt 4645 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: sge0f1o 39275 |
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