Theorem elrnmpo 5884

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)
elrnmpo.1	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
elrnmpo	⊢ (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 elrnmpo

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	rnmpo 5881	. . 3 ⊢ ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶}
3	2	eleq2i 2206	. 2 ⊢ (𝐷 ∈ ran 𝐹 ↔ 𝐷 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶})
4		elrnmpo.1	. . . . . 6 ⊢ 𝐶 ∈ V
5		eleq1 2202	. . . . . 6 ⊢ (𝐷 = 𝐶 → (𝐷 ∈ V ↔ 𝐶 ∈ V))
6	4, 5	mpbiri 167	. . . . 5 ⊢ (𝐷 = 𝐶 → 𝐷 ∈ V)
7	6	rexlimivw 2545	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝐷 = 𝐶 → 𝐷 ∈ V)
8	7	rexlimivw 2545	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶 → 𝐷 ∈ V)
9		eqeq1 2146	. . . 4 ⊢ (𝑧 = 𝐷 → (𝑧 = 𝐶 ↔ 𝐷 = 𝐶))
10	9	2rexbidv 2460	. . 3 ⊢ (𝑧 = 𝐷 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶))
11	8, 10	elab3 2836	. 2 ⊢ (𝐷 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶)
12	3, 11	bitri 183	1 ⊢ (𝐷 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶)

Syntax hints:   ↔ wb 104   = wceq 1331   ∈ wcel 1480  {cab 2125  ∃wrex 2417  Vcvv 2686  ran crn 4540   ∈ cmpo 5776

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-cnv 4547  df-dm 4549  df-rn 4550  df-oprab 5778  df-mpo 5779

This theorem is referenced by: (None)