Theorem elrnmpo 7536

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 7533	. . 3 ⊢ ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶}
3	2	eleq2i 2857	. 2 ⊢ (𝐷 ∈ ran 𝐹 ↔ 𝐷 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶})
4		elrnmpo.1	. . . . . 6 ⊢ 𝐶 ∈ V
5		eleq1 2853	. . . . . 6 ⊢ (𝐷 = 𝐶 → (𝐷 ∈ V ↔ 𝐶 ∈ V))
6	4, 5	mpbiri 261	. . . . 5 ⊢ (𝐷 = 𝐶 → 𝐷 ∈ V)
7	6	rexlimivw 3162	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝐷 = 𝐶 → 𝐷 ∈ V)
8	7	rexlimivw 3162	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶 → 𝐷 ∈ V)
9		eqeq1 2769	. . . 4 ⊢ (𝑧 = 𝐷 → (𝑧 = 𝐶 ↔ 𝐷 = 𝐶))
10	9	2rexbidv 3230	. . 3 ⊢ (𝑧 = 𝐷 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶))
11	8, 10	elab3 3648	. 2 ⊢ (𝐷 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶)
12	3, 11	bitri 278	1 ⊢ (𝐷 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶)

Syntax hints:   ↔ wb 209   = wceq 1563   ∈ wcel 2145  {cab 2743  ∃wrex 3089  Vcvv 3457  ran crn 5653   ∈ cmpo 7402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-cnv 5660  df-dm 5662  df-rn 5663  df-oprab 7404  df-mpo 7405

This theorem is referenced by:  qexALT  12979  lsmelvalx  19701  efgtlen  19787  frgpnabllem1  19934  fmucndlem  24408  mbfimaopnlem  25775  tglnunirn  28775  isplng  29008  tpr2rico  34219  mbfmco2  34572  br2base  34576  dya2icobrsiga  34583  dya2iocnrect  34588  dya2iocucvr  34591  sxbrsigalem2  34593  cntotbnd  38307  eldiophb  43350  elicores  46107  volicorescl  47125