Theorem fnasrng 5857

Description: A function expressed as the range of another function. (Contributed by Jim Kingdon, 9-Jan-2019.)

Assertion

Ref	Expression
fnasrng	⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) = ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩))

Proof of Theorem fnasrng

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfmptg 5856	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩})
2		eqid 2232	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩) = (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩)
3	2	rnmpt 5004	. . . 4 ⊢ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝐵⟩}
4		velsn 3705	. . . . . 6 ⊢ (𝑦 ∈ {⟨𝑥, 𝐵⟩} ↔ 𝑦 = ⟨𝑥, 𝐵⟩)
5	4	rexbii 2549	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩} ↔ ∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝐵⟩)
6	5	abbii 2348	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝐵⟩}
7	3, 6	eqtr4i 2256	. . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}}
8		df-iun 3992	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}}
9	7, 8	eqtr4i 2256	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩) = ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩}
10	1, 9	eqtr4di 2283	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) = ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩))

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203  {cab 2218  ∀wral 2520  ∃wrex 2521  {csn 3688  ⟨cop 3691  ∪ ciun 3990   ↦ cmpt 4170  ran crn 4749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358

This theorem is referenced by:  resfunexg  5904