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Mathbox for Glauco Siliprandi |
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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 5910, 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 2736 | . . 3 ⊢ 𝐶 = 𝐶 | |
2 | elrnmpt1sf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
3 | 2, 2 | nfeq 2918 | . . . 4 ⊢ Ⅎ𝑥 𝐶 = 𝐶 |
4 | elrnmpt1sf.3 | . . . . 5 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) | |
5 | 4 | eqeq2d 2747 | . . . 4 ⊢ (𝑥 = 𝐷 → (𝐶 = 𝐵 ↔ 𝐶 = 𝐶)) |
6 | 3, 5 | rspce 3568 | . . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 = 𝐶) → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) |
7 | 1, 6 | mpan2 689 | . 2 ⊢ (𝐷 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) |
8 | elrnmpt1sf.2 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
9 | 2, 8 | elrnmptf 43335 | . . 3 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) |
10 | 9 | biimparc 480 | . 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 396 = wceq 1541 ∈ wcel 2106 Ⅎwnfc 2885 ∃wrex 3071 ↦ cmpt 5186 ran crn 5632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5166 df-mpt 5187 df-cnv 5639 df-dm 5641 df-rn 5642 |
This theorem is referenced by: sge0f1o 44555 |
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