Theorem fnasrn 5887

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.)

Hypothesis

Ref	Expression
dfmpt.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
fnasrn	⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩)

Proof of Theorem fnasrn

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfmpt.1	. . 3 ⊢ 𝐵 ∈ V
2	1	dfmpt 5886	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩}
3		eqid 2238	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩) = (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩)
4	3	rnmpt 5030	. . . 4 ⊢ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝐵⟩}
5		velsn 3726	. . . . . 6 ⊢ (𝑦 ∈ {⟨𝑥, 𝐵⟩} ↔ 𝑦 = ⟨𝑥, 𝐵⟩)
6	5	rexbii 2557	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩} ↔ ∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝐵⟩)
7	6	abbii 2354	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝐵⟩}
8	4, 7	eqtr4i 2262	. . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}}
9		df-iun 4014	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}}
10	8, 9	eqtr4i 2262	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩) = ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩}
11	2, 10	eqtr4i 2262	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩)

This proof depends on syntax axioms:   = wceq 1402   ∈ wcel 2209  {cab 2224  ∃wrex 2529  Vcvv 2821  {csn 3709  ⟨cop 3712  ∪ ciun 4012   ↦ cmpt 4192  ran crn 4775

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384

This theorem is used by:  idref  5962