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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 5606, 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 | nfcv 2970 | . . . 4 ⊢ Ⅎ𝑥𝑦 | |
2 | elrnmptf.1 | . . . 4 ⊢ Ⅎ𝑥𝐶 | |
3 | 1, 2 | nfeq 2982 | . . 3 ⊢ Ⅎ𝑥 𝑦 = 𝐶 |
4 | eqeq1 2830 | . . 3 ⊢ (𝑦 = 𝐶 → (𝑦 = 𝐵 ↔ 𝐶 = 𝐵)) | |
5 | 3, 4 | rexbid 3262 | . 2 ⊢ (𝑦 = 𝐶 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) |
6 | elrnmptf.2 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
7 | 6 | rnmpt 5605 | . 2 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
8 | 5, 7 | elab2g 3575 | 1 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1658 ∈ wcel 2166 Ⅎwnfc 2957 ∃wrex 3119 ↦ cmpt 4953 ran crn 5344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4875 df-opab 4937 df-mpt 4954 df-cnv 5351 df-dm 5353 df-rn 5354 |
This theorem is referenced by: elrnmpt1sf 40185 |
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