Theorem fnasrn 6298

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 6297	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩}
3		eqid 2605	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩) = (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩)
4	3	rnmpt 5275	. . . 4 ⊢ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝐵⟩}
5		velsn 4136	. . . . . 6 ⊢ (𝑦 ∈ {⟨𝑥, 𝐵⟩} ↔ 𝑦 = ⟨𝑥, 𝐵⟩)
6	5	rexbii 3018	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩} ↔ ∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝐵⟩)
7	6	abbii 2721	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝐵⟩}
8	4, 7	eqtr4i 2630	. . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}}
9		df-iun 4447	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}}
10	8, 9	eqtr4i 2630	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩) = ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩}
11	2, 10	eqtr4i 2630	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩)

Syntax hints:   = wceq 1474   ∈ wcel 1975  {cab 2591  ∃wrex 2892  Vcvv 3168  {csn 4120  ⟨cop 4126  ∪ ciun 4445   ↦ cmpt 4633  ran crn 5025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793

This theorem is referenced by:  resfunexg  6358  idref  6377  gruf  9485