Theorem fnasrng 5676

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 5675	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩})
2		eqid 2170	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩) = (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩)
3	2	rnmpt 4859	. . . 4 ⊢ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝐵⟩}
4		velsn 3600	. . . . . 6 ⊢ (𝑦 ∈ {⟨𝑥, 𝐵⟩} ↔ 𝑦 = ⟨𝑥, 𝐵⟩)
5	4	rexbii 2477	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩} ↔ ∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝐵⟩)
6	5	abbii 2286	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝐵⟩}
7	3, 6	eqtr4i 2194	. . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}}
8		df-iun 3875	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}}
9	7, 8	eqtr4i 2194	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩) = ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩}
10	1, 9	eqtr4di 2221	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) = ran (𝑥 ∈ 𝐴 ↦ ⟨𝑥, 𝐵⟩))

Syntax hints:   → wi 4   = wceq 1348   ∈ wcel 2141  {cab 2156  ∀wral 2448  ∃wrex 2449  {csn 3583  ⟨cop 3586  ∪ ciun 3873   ↦ cmpt 4050  ran crn 4612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205

This theorem is referenced by:  resfunexg  5717