Theorem fnasrng 5286

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 5285	. 2 ⊢ (∀x ∈ A B ∈ 𝑉 → (x ∈ A ↦ B) = ∪ x ∈ A {⟨x, B⟩})
2		eqid 2037	. . . . 5 ⊢ (x ∈ A ↦ ⟨x, B⟩) = (x ∈ A ↦ ⟨x, B⟩)
3	2	rnmpt 4525	. . . 4 ⊢ ran (x ∈ A ↦ ⟨x, B⟩) = {y ∣ ∃x ∈ A y = ⟨x, B⟩}
4		elsn 3382	. . . . . 6 ⊢ (y ∈ {⟨x, B⟩} ↔ y = ⟨x, B⟩)
5	4	rexbii 2325	. . . . 5 ⊢ (∃x ∈ A y ∈ {⟨x, B⟩} ↔ ∃x ∈ A y = ⟨x, B⟩)
6	5	abbii 2150	. . . 4 ⊢ {y ∣ ∃x ∈ A y ∈ {⟨x, B⟩}} = {y ∣ ∃x ∈ A y = ⟨x, B⟩}
7	3, 6	eqtr4i 2060	. . 3 ⊢ ran (x ∈ A ↦ ⟨x, B⟩) = {y ∣ ∃x ∈ A y ∈ {⟨x, B⟩}}
8		df-iun 3650	. . 3 ⊢ ∪ x ∈ A {⟨x, B⟩} = {y ∣ ∃x ∈ A y ∈ {⟨x, B⟩}}
9	7, 8	eqtr4i 2060	. 2 ⊢ ran (x ∈ A ↦ ⟨x, B⟩) = ∪ x ∈ A {⟨x, B⟩}
10	1, 9	syl6eqr 2087	1 ⊢ (∀x ∈ A B ∈ 𝑉 → (x ∈ A ↦ B) = ran (x ∈ A ↦ ⟨x, B⟩))

Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  {cab 2023  ∀wral 2300  ∃wrex 2301  {csn 3367  ⟨cop 3370  ∪ ciun 3648   ↦ cmpt 3809  ran crn 4289

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852

This theorem is referenced by:  resfunexg  5325