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Mirrors > Home > MPE Home > Th. List > fnasrn | Structured version Visualization version GIF version |
Description: A function expressed as the range of another function. (Contributed by Mario Carneiro, 22-Jun-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
dfmpt.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
fnasrn | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfmpt.1 | . . 3 ⊢ 𝐵 ∈ V | |
2 | 1 | dfmpt 6908 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉} |
3 | eqid 2823 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) = (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) | |
4 | 3 | rnmpt 5829 | . . . 4 ⊢ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝐵〉} |
5 | velsn 4585 | . . . . . 6 ⊢ (𝑦 ∈ {〈𝑥, 𝐵〉} ↔ 𝑦 = 〈𝑥, 𝐵〉) | |
6 | 5 | rexbii 3249 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {〈𝑥, 𝐵〉} ↔ ∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝐵〉) |
7 | 6 | abbii 2888 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {〈𝑥, 𝐵〉}} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝐵〉} |
8 | 4, 7 | eqtr4i 2849 | . . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {〈𝑥, 𝐵〉}} |
9 | df-iun 4923 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {〈𝑥, 𝐵〉}} | |
10 | 8, 9 | eqtr4i 2849 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) = ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉} |
11 | 2, 10 | eqtr4i 2849 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 {cab 2801 ∃wrex 3141 Vcvv 3496 {csn 4569 〈cop 4575 ∪ ciun 4921 ↦ cmpt 5148 ran crn 5558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 |
This theorem is referenced by: idref 6910 resfunexg 6980 gruf 10235 |
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