Theorem elrnmpt2 5556

Description: Membership in the range of an operation class abstraction. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypotheses

Ref	Expression
rngop.1	⊢ 𝐹 = (x ∈ A, y ∈ B ↦ 𝐶)
elrnmpt2.1	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
elrnmpt2	⊢ (𝐷 ∈ ran 𝐹 ↔ ∃x ∈ A ∃y ∈ B 𝐷 = 𝐶)

Distinct variable groups:   y,A   x,y,𝐷

Allowed substitution hints:   A(x)   B(x,y)   𝐶(x,y)   𝐹(x,y)

Proof of Theorem elrnmpt2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngop.1	. . . 4 ⊢ 𝐹 = (x ∈ A, y ∈ B ↦ 𝐶)
2	1	rnmpt2 5553	. . 3 ⊢ ran 𝐹 = {z ∣ ∃x ∈ A ∃y ∈ B z = 𝐶}
3	2	eleq2i 2101	. 2 ⊢ (𝐷 ∈ ran 𝐹 ↔ 𝐷 ∈ {z ∣ ∃x ∈ A ∃y ∈ B z = 𝐶})
4		elrnmpt2.1	. . . . . 6 ⊢ 𝐶 ∈ V
5		eleq1 2097	. . . . . 6 ⊢ (𝐷 = 𝐶 → (𝐷 ∈ V ↔ 𝐶 ∈ V))
6	4, 5	mpbiri 157	. . . . 5 ⊢ (𝐷 = 𝐶 → 𝐷 ∈ V)
7	6	rexlimivw 2423	. . . 4 ⊢ (∃y ∈ B 𝐷 = 𝐶 → 𝐷 ∈ V)
8	7	rexlimivw 2423	. . 3 ⊢ (∃x ∈ A ∃y ∈ B 𝐷 = 𝐶 → 𝐷 ∈ V)
9		eqeq1 2043	. . . 4 ⊢ (z = 𝐷 → (z = 𝐶 ↔ 𝐷 = 𝐶))
10	9	2rexbidv 2343	. . 3 ⊢ (z = 𝐷 → (∃x ∈ A ∃y ∈ B z = 𝐶 ↔ ∃x ∈ A ∃y ∈ B 𝐷 = 𝐶))
11	8, 10	elab3 2688	. 2 ⊢ (𝐷 ∈ {z ∣ ∃x ∈ A ∃y ∈ B z = 𝐶} ↔ ∃x ∈ A ∃y ∈ B 𝐷 = 𝐶)
12	3, 11	bitri 173	1 ⊢ (𝐷 ∈ ran 𝐹 ↔ ∃x ∈ A ∃y ∈ B 𝐷 = 𝐶)

Syntax hints:   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {cab 2023  ∃wrex 2301  Vcvv 2551  ran crn 4289   ↦ cmpt2 5457

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-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-cnv 4296  df-dm 4298  df-rn 4299  df-oprab 5459  df-mpt2 5460

This theorem is referenced by: (None)