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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elrnmptf | Structured version Visualization version GIF version |
Description: The range of a function in maps-to notation. Same as elrnmpt 5950, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
elrnmptf.1 | ⊢ Ⅎ𝑥𝐶 |
elrnmptf.2 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
elrnmptf | ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrnmptf.1 | . . . 4 ⊢ Ⅎ𝑥𝐶 | |
2 | 1 | nfeq2 2921 | . . 3 ⊢ Ⅎ𝑥 𝑦 = 𝐶 |
3 | eqeq1 2737 | . . 3 ⊢ (𝑦 = 𝐶 → (𝑦 = 𝐵 ↔ 𝐶 = 𝐵)) | |
4 | 2, 3 | rexbid 3272 | . 2 ⊢ (𝑦 = 𝐶 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) |
5 | elrnmptf.2 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
6 | 5 | rnmpt 5949 | . 2 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
7 | 4, 6 | elab2g 3668 | 1 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 Ⅎwnfc 2884 ∃wrex 3071 ↦ cmpt 5227 ran crn 5673 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5145 df-opab 5207 df-mpt 5228 df-cnv 5680 df-dm 5682 df-rn 5683 |
This theorem is referenced by: elrnmpt1sf 43758 |
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