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| 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 7163 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉} | 
| 3 | eqid 2736 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) = (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) | |
| 4 | 3 | rnmpt 5967 | . . . 4 ⊢ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝐵〉} | 
| 5 | velsn 4641 | . . . . . 6 ⊢ (𝑦 ∈ {〈𝑥, 𝐵〉} ↔ 𝑦 = 〈𝑥, 𝐵〉) | |
| 6 | 5 | rexbii 3093 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {〈𝑥, 𝐵〉} ↔ ∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝐵〉) | 
| 7 | 6 | abbii 2808 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {〈𝑥, 𝐵〉}} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝐵〉} | 
| 8 | 4, 7 | eqtr4i 2767 | . . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {〈𝑥, 𝐵〉}} | 
| 9 | df-iun 4992 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {〈𝑥, 𝐵〉}} | |
| 10 | 8, 9 | eqtr4i 2767 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) = ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉} | 
| 11 | 2, 10 | eqtr4i 2767 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∈ wcel 2107 {cab 2713 ∃wrex 3069 Vcvv 3479 {csn 4625 〈cop 4631 ∪ ciun 4990 ↦ cmpt 5224 ran crn 5685 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 | 
| This theorem is referenced by: idref 7165 resfunexg 7236 gruf 10852 | 
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